The ideal group size is 3 or 4. Fewer than that, and the group dynamics aren't very interesting and the collective skills of the characters may not be large enought to cover all the problems the GM will throw at them. More than 4 players can be very wearisome for the GM and frustrating for the players because nobody feels like they're getting any stage time. The following are some guidelines that can hopefully help the GM manage a large group.
There are some things you can do to save everyone's sanity before the adventure even starts.
This is the "ounce of prevention" guideline: don't invite a big group in the first place and you won't have to deal with the problems. This is not always possible though, you might have a pickup game that just starts when a whole lot (and I mean a whole lot) of gamers are gathered. (There are 2d6 gamers in the room. Roll to see how many... Doh! Boxcars!) Alternatively, if you have some players that occasionally don't make it, you might invite, say, 6-8 people expecting only half of them to show up. Lucky you, they all show up.
If you have a large number of people in your gaming circle, you might try to get them to buy into the "Large Groups Equals Bad" idea and invite them to politely bow out of one session for the greater good of the rest of the group. Hopefully, other members of the gaming circle can reciprocate to the increased enjoyment of all involved.
When everyone arrives, character sheets should be completed, earned character points should be spent and equipment bought as desired, the adventure should be entirely written, and any house rules / new rules should already be decided on. This is an ideal scenario and it never happens this way. At the very least, you should be able to have the character sheets and the adventure done. You can even have a pre-adventure session where you take care of all these details.
This guideline should be observed at all times anyway, but it's doubly important when you have a large group.
Here's a common situation: The GM introduces an important NPC that the PCs need to talk to to get information. All the PCs immediately swarm around the hapless NPC, buffeting him with questions, snappy patter, and overblown, hammed-up roleplaying. Sadly, the one who takes the brunt of all this abuse is the GM.
So, before the adventure begins, it's a good idea to have the players select who will be the point-man for particular tasks and encounters. Some of these tasks are rather self-selecting: the best-looking / most famous will probably be the one who does the talking, the theif will do all the lockpicking & breaking and entering, the cleric will go around healing everyone after combat, and so forth. Sometimes there isn't a clear candidate for a job and it's for these cases that a decision needs to be made.
Naturally, it is impossible to determine all of the situations that will arise in a given adventure, but, as the GM, you should have a pretty good idea of the more likely situations that will occur.
Once assignments have been made about who will do what, write them down so that, when situation X arises, you can address directly the player who was selected for situation X, rather than just throwing it out there for the group to argue about.
Pick someone, or have the players pick someone, who will be the spokesman for the group. This way, the GM is sheilded from the plan-hatching, discussion, and bickering of the players. All the GM has to do is sit still and wait to hear what the Party Leader has to say after the group is done wrangling.
This has a lot of crossover with the "Divide the Labor" principle above. After the Party Leader is selected, the GM can then ask him to divide the labor for you.
Picking a good Party Leader is a balancing act. On the one hand, you don't want to pick someone who's going to be a complete dictator because the other players will feel like they're always getting steamrolled by him. On the other hand, if you pick a Party Leader who always believes in getting consensus, the game could drag out for hours as the group discusses every little approach and viewpoint before making a decision. (See also "It Ain't a Chess Game" below -- Only on Star Trek: The Next Generation are the characters able to defeat their enemy by having long, boardroom discussions.) A party leader who can win the party's favor and rally the party to come up with a quick plan-of-action is a rare and worthy find.
Because you can't deal with everyone at once, it can help you prepare some reading / study materials that pertain to the adventure (such as a map or newspaper). This way, while you're talking with some of the players, the study materials can keep the rest of them busy.
If you have a trial or test of some kind that requires teamwork, it helps all the players get involved and feel validated; everybody's getting stage time, everybody's getting a chance to use their special skills and powers. Preparing such a scene typically requires some up-front preparation by the GM with all the character sheets in hand.
Explain that if someone has to leave early, it's not a problem. Better yet, find out if anyone needs to leave early and do a lot of stuff with them toward the beginning of the game. This way, when they need to go, they won't feel like they've been short-shrifted because they had to share timeslices with everyone else but couldn't stay as long.
The members of large parties frequently discover that with so many people in the group, they need to compete for the attention of the game master. Humans being the self-organizing creatures that they are, it will dawn on them that they won't have to share time with as many people if they split off into a smaller group. Players with particular loner / showoff / maverick tendencies will prefer the split off group to have a size of one: just them, no one else.
Alternatively, you, the GM, might determine that it is easier to manage 2 groups of 4 than 1 group of 8. A logical and valid conclusion, this nevertheless has it's downsides. Whichever group you're not dealing with is likely to get bored or hear things they shouldn't know about. It can also be difficult to try to sync-up everyone in terms of game-time when / if they re-unite.
And then there are those occasions when half the players arrive at noon and the other half show up an hour later. In these sort of situations, a split-up is sometimes unavoidable.
Split up parties are awkward to manage at best and can be really onerous when you have a large party that gets split into 3 or more groups. Here's some strategies that can help.
Yet another "ounce of prevention", you can usually prevent a split-up before it even happens simply by saying "Are you guys sure you want to split up? I mean, I don't think that would be such a good idea." Chances are, the players will believe you.
You can even make a declaration to this effect before the adventure begins: "Okay you guys, there's a lot of you here, but I really think you should stick together. Just a word to the wise, here."
This should be your first goal as a GM after (or even before) the party splits up: figure out how and when to get the seperate groups back together. You may not do it right away, but you'll probably want to do it eventually.
Remember that you have awesome, cosmic powers at your disposal for getting the group back together. Swirly portals can mysteriously appear out of nowhere and transport your characters to a central meeting place. A cellar door can surprisingly lead to a room in the attic where everyone else is assembled. PCs can be carried off by giant Tse-Tse flies to the other side of the mountain where the rest of the party is gathered. Be creative. Remeber, you're Game Operations Director (G.O.D.) here.
Don't allow any single group to consume all of the game time with their own activities; it's not fair to the rest of the players and they'll feel cheated and neglected. Switching frequently between the groups gives everybody a turn on stage and prevents any one group from feeling short-changed.
Specific things to avoid with a splinter group are combat and lengthy contests (like card-playing sessions) because these can take a lot of time. If the splinter group is dead-set on getting in a fight, stop the action right before the first blow is thrown and switch to the other group. Then come up with a contrived way for the other group to re-join them and join the melee.
Try to have some other activity nearby that can occupy the "waiting" party so that they're not getting bored or hearing things they shouldn't. Good examples of extra-curricular activities are a TV, a Nintendo, or another game (a board game, a card game, etc.). Some players might even be content to read the rulebook and familiarize themselves with the rules better or make a new character. Artistically inclined players might draw pictures of their character or the group. The advantage with rulebook and artwork activities is that they keep the players' attention on the game, but keep them distracted for a little while as well.
The extra-curricular activity (or activities) should be something that can be picked up easily and droped quickly when it's time for them to come back. If it so happens that they just can't break themselves away from the extra-curricular activity, so much the better; it's one less person you have to deal with. (Unless, of course, you really want them to come back and finish the adventure you prepared...)
This strategy is most useful when you have just one person (or maybe two) in a split-up party. Rather than making everyone else stall while you engage in a lengthy one-on-one with the soloist, tell him to write a note describing what he's doing and where he's going. This will keep the soloist busy while you carry on with the larger group. Then, later on, you can read the note, respond, and either re-unite him with the party or repeat the note-passing again.
You could even ask the soloist to write a note, leave it for you, and then go engage in an extra-curricular activity, letting him know that you'll get back to him when you're ready for him.
If people want to split up, you can use this as an occasion to reduce the total number of players you have to deal with. The splinter group can fall in a hole, get knocked unconcious, get hauled off to prison, get trapped in the mine shaft, get sprayed with sleeping gas, or any number of other schemes that will (temporarily) remove them from play. These removed players can then go engage in an extra-curricular activity until they wake up, get rescued, etc.
Killing off the players just to reduce your stress is probably not a good idea. Chances are, the resulting bad ju-ju's will just increase your stress.
This approach is recommended here as a suggestion for handling netrunning, a notoriously exclusive activity. Basically, you try to do the solo group either before or after everyone else. Sometimes this is feasible, sometimes it's not.
In real life, combat is fast-paced and quickly resolved. In gaming, combat is the most time-consuming, number-crunching aspect of the system. One thing the real-life and game versions of combat have in common is that they are both fatiguing. The more people you have in the group, the more fatiguing it will be for the everyone involved but especially for the GM.
Write down stats, charts and tracking tables before combat begins. If you can, write these up before the adventure begins, so as to consume less game time. There are all kinds of play aids that you can use to help you manage all the number crunching, but they'll only help if you use them.
It might also save you time to use a little mini-whiteboard and a dry-erase pen to keep track of current hit points, etc. rather than a pen or pencil, cuz you can erase faster. (On the downside, one bad swipe and you lose all your stats...)
Some systems are designed with very simple combat rules and already go pretty quickly. Some game systems pride themselves on their "realism", much to the agony of the players involved. Numerous approaches are available for simplifying an involved combat system, some of which might even be published in "official" suppliments.
One guideline for simplifying combat is to turn some rolls into static numbers. For example, in some games an initiative roll is called for on every round. GURPS instead uses a pre-calculated 'Move' value to determine the order of turns in a combat round. This eliminates one roll per player, but more importantly it eliminates a load of bookwork that the GM would otherwise have to do, rolling and writing down the the turn order every round.
Another guideline for simplifying combad is to make some rolls modifiers to a previous roll. For example, a round might require a attack roll, a defense roll, and a damage roll. You can combine the first two rolls into one by subtracting the defense stat from the attack roll. (This strategy is especially useful if both combatants have very high stats.) The damage could then be determined by the success spread of the attackers roll, but you would probably want to roll seperately for the damage because this is a sensetive area and damage might not even be called for on a miss.
Some RPGs have rules for "teaming up" or mass-combat that simply combines the stats of the players on the same team into one big roll. This way, you can resolve huge, barroom-size fights with a few rolls. The Compendium II "Combat and Campaigns" has some rules for resolving Mass Combat fairly quickly.
For a simplified, quick-and-dirty combat system, see: Very Basic Melee Combat "Really Simple Shortcuts for Really Quick Battles" by Steffan O'Sullivan.
Another rule of thumb: Don't use the Advanced Combat Rules in large groups!
It's bad enought that you have a lot of players to deal with, don't make life worse by having a lot of opponents to deal with as well. Instead of lots of little henchmen, make fewer, tougher opponents. It will reduce the bookwork you have to do.
Another way to reduce the number of opponents in the middle of a combat session is to have them drop to the ground after one hit (hey, maybe they have the 'Low Pain Threshold' disadvantage and one hit takes them out), or have them wimp out and run away, possibly even before they take a hit (hey, maybe they're easily frightened).
You know, you don't have to engage in huge 20+ person combat. There are ways to reduce the number of combatants. In olden time, it was customary for two armies to select their best man and then just the two of them would duke it out to resolve the battle. You could do a similar sort of thing.
Combat is fast and furious. It should feel that way to the players involved. If they can't make a decision about what they're going to do next then they're paralyzed / stunned for the round and it's the next person's turn.
Most gaming sessions have A) some roleplaying and B) some dice rolling. A good game will mix the two so as to keep things interesting and entertaining. Combat sessions usually consist of lots of dice rolling, but not a lot of role-playing. As a result, the level of entertainment and joviality drops. In a large group where everyone has to take turns, this can lead to bored players and lots of dice spinning.
Encourage players to keep the banter up even durring combat. They can play their quirks, spout their catch phrases, pantomime actions, and otherwise crack wise. This should keep the humor level up and keep people laughing even when there's lots of number crunching going on.
Let's say the party is in a bar and spoiling for a fight, so they start one. This fight does nothing to advance the story and is completely irrelevant to the larger picture. Why then, should you waste time with a lot of rolling? Don't bother. Per the "Encourage Roleplaying in Combat" section above, just have everyone narrate what they're doing and you can narrate the responses. If you want to do one or two rolls just to account for dumb luck or happenchance, fine, but don't waste everyone's time trying to number-crunch the whole thing if it isn't going to matter anyway.