What is in a round?
Before you can look at what happens in a round you need to know what it is. Most systems have things broken down into three different terms: Actions, Rounds, and Turns. While most modern systems are generally consistent with what those means, if you're dealing with an older system you should double check the terminology, as some (like AD&D 2nd edition) use a Turn to mean 10 rounds instead of how I define it here.
Action: An action is what a player's specific go around is made up of. Most modern systems have this broken down into multiple things but some do not. Attacking an opponent is an action, running 20' is an action. Reloading your gun is an action. How many actions you have on your go depends on the system and your character, but this is generally the quickest thing to resolve in an over-all round and rarely has a given amount of time specified to it.
Round: Generally speaking a round is an entire run through the initiative order. From top to bottom or vice versa. A round is usually what is given a time on. This is also generally what we are considering when it comes to different timers, their uses and restrictions.
Turn: In most systems now a days, a Turn is an individual players turn in a round. This doesn't mean anything inside the round for actual timing, so much as the artificial ordering for when you get to resolve your chunk of actions. However, as said above, in some systems a Turn is actually a collection of rounds and used as a different timer
Now that we have the basic terms out of the way, I get to tell you the fun part. While knowing 'Action' and 'Turn' is important, the most important thing, for this discussion at least, is a round and what it means. That being said, lets look at some of the more common round times and their uses.
The 1 Minute Round
The 1 minute round is mostly a thing of the past now-a-days I believe, but was probably most famously featured in Advanced Dungeons and Dragons. It may have also been in Dungeons and Dragons, but it's been too long for me to remember. This way of doing things has a lot of good things and a lot of bad things about it. Sadly, one of the bad things is how it makes the players feel if the GM doesn't step it up a bit with the descriptions of what is going on. At the same time however, a 1 minute round gives you longer fights meaning things like the cops showing up after just 3 rounds (in which any gamer can tell you a LOT can happen) is not as unusual or strange as it is with a shorter round time.
With a 1 minute round, a player's action and their one to three attacks don't represent their only attacks but rather how they do over all during the minute, or their ability to exploit the openings presented to them during the minute. This is where the GM needs to step things up with description in between rounds, as if just left flat, it feels like the player can only swing his sword three times over a minute, and lets face it, even an untrained person can do more than that. However, when you take into account that it is three openings, and your GM describes the fight well, that one minute round can be epic combat with back and forth sword play, and being chased around the whole area in the fight. Done right, you get long fights like out of the movies, done wrong you have bored and confused players wondering why they can't do more in just a minute.
That lack of ability to do things though isn't only when you want to attack. What about if someone is at a dead run? Movement rates need to be done to make sense for a minute of all out movement. The same with other actions that can happen during a combat round that aren't directly related to combat. This is where things get really tricky, and why I feel the 1 minute round has more or less become a thing of the past. For all that it can do well, too much that happens around the actual direct combat application can be muddled.
The 6 Second Round
The six second round seems to be the most common round timer that is going around now a days. Then again, it's easy to see why, it does after all make keeping track of time easy. 10 rounds is a minute. 600 rounds is an hour. It also hits a nice balance point between allowing players to have a variety of things they can do in one round, while not falling into the problem too much of "what do you mean I can only swing once in a minute?" That being said, it doesn't fix the problem completely.
Still, it is something to consider for a game you are designing, or for modifying your own round timers to. The ability to easily keep track of time with it, while only sacrificing a little plausibility makes it a good place to go to. The GM here should still take the time to describe actions as being longer than the simple "attack, hit" mechanic that is actually being rolled out. There is more time than is actually taken by a player's action to be explained. Though some of that can be explained by getting the players more involved, have them explain how they go into their action. Reward good and plausible descriptions to get them into it. It makes the combat more enjoyable for everyone that way.
The 3 Second Round
The three second round has some of the strengths of the six second round, while resolving a lot of the problems the longer round times have. A three second round, ironically, suffers the opposite problem of the six second round when it starts coming to player actions. In games where a higher level means more attacks, you start to get into the 'fantastical' speed level for characters. Able to easily dispatch not one or two but four or more people in a single round. That being said, this is easily fixed by keeping the number of attacks a person can do in a round.
Communication is strangely easier here, though needing to be kept under control by the GM. It's easy to say one sentence in three seconds, provided it's not too long. However, a tendency for conversations to start could mean you need to keep what players say under control. If you don't mind that, have at, but those looking to keep things more 'realistic' should be aware of that.
The three second rounds other problem comes in with larger games and dramatic applications. A LOT can happen in a round, and int he real world resolving one round of combat can take a LOT of time. Waiting one hour to resolve 3 seconds of in game time can be very frustrating, and lead to boredom. Especially when the GM is curtailing what can and can't be said for plausibility.
The 1 second round
This is the round timer used in games like GURPS, and while it does have it's strengths I personally don't like it. While a lot can happen in a second, the constraints that it can bring are hard to deal with as a player in my opinion. Conversation more or less needs to stop, as you can really only say 1 word in a second, meaning any attempt for meaningful banter or orders is going to be broken up over 5-10 minutes in the real world as you wait for your turn to say something else. Also, while games using 1 second round timers usually compensate for this by allowing you to either move OR attack OR aim OR ready a weapon, there are ways around those too usually, and even with such constraints a LOT can still happen in 3 rounds or less. Finally, unless you're looking for this, it can be jarring just how much players can do in those 1 second rounds when you get to the higher levels. Now, while sure you're at higher levels so it's to be expected, when you see someone draw two guns, empty their clips into 4 people, discard the guns, and draw two more all in 1 second. It's a little weird.
That being said, the 1 second round does have it's uses. Especially in close combat where you really are dealing with second by second exchanges. Watch The Bourne Identity, the scene where Bourne gets attacked in his apartment in paris, systems like GURPS with their one second round times can accurately recreate that. At the same time though, that same scene which takes.
So, which do I use then?
Honestly, that really depends on the game, and more to the point, the scenario in question. Some things are intrinsically tied into the round timer in a game, and you need to pay attention to those. That said, depending on the situation different round timers may be good to go to. Is your duelist PC having his final confrontation with the guy who killed his dad? Maybe switch it to a 1 minute round timer for the fight, and go nuts describing the interplay as those attack rolls they make are just the openings they get. A more serious, high drama moment? Drop down to a slower time frame like 1 or 3 second rounds and make them make each second count. You need to be careful when you do this, most systems aren't designed to be used with different round timers too well while feeling natural. But if your players trust you, it can add a lot to the game experience for things.
That being said, the round timer you should use in general I feel really depends on the group size. If you have 3-4 players (or more importantly, the game is designed for 3-4 players and a gm) a 3 second round timer I feel works rather well. Combat goes smoothly, minutes aren't too hard to keep track of, and a decent amount can be done in 3 seconds without feeling like too much. If your game has 4-6 players, 6 or 10 second rounds might be the way to go. It gives players a bit more leeway for conversation in their rounds, and while less may be happening per character, more time is passing faster. For more than 6 players, you may want to consider going up to a minute per round. The length of time OOC that you are spending on each round is only going longer, and dropping the round to a minute per while meaning you need to describe more, also allows you to with description make players feel like more is going on.
Playing with round timers though is something you should only do if you and your group know and trust each other with this sort of thing. If you guys like going "by the book" then go by the book. If you are more loose and wild, this is more for you than the other crowd. For designing a game though, you need to ask yourself what are you trading off, and what are you gaining, with wherever you set your round to.
Things to keep in mind
When setting a round timer for a game, you need to keep things in mind.
Attacks - Is the player rolling for every individual attack that he is doing over the course of the round? Or is it for how well he does against that opponent for the round? If you want him rolling for the individual attacks, then you want a shorter round timer. If it's for the openings he finds, or the over all impact, a longer round time.
Movement - While movement doesn't impact the round timer, movement rates are impacted by it. People move further in a minute than 6 seconds, it's just a simple fact. So you want to make sure your movement rates are reflective. Keep in mind, most PCs are fit and capable, a 5 second 40 yard dash is probably the average for the level of fit most PCs will have. Meaning that in a full on sprint they can cover about 24' in one second. That is a full on sprint however, not focusing on anything else. A useful guide for 1 to 10 second rounds, but less so for a minute as you rarely sprint all out for a minute or more.
Other little actions - How much of your turn does it take to do things like open a door? Reload your pistol? Grab a knife from it's boot sheathe? Now while specific and special training, or the feel of the world, could dictate these things going faster, generally speaking each of these is a multi-second affair. Your hand has to go down, from wherever it is, to where the item is stored. Get a grip, pull it out, and ready it. For a 3 second round or less, you want to seriously consider these things taking a players turn without special abilities or training. For six seconds or more, a part of the turn. At a minute, it shouldn't even really factor, as a few seconds out of 60 isn't all that big of a deal now really.
Longer actions and arriving help - No matter how long you set your round timer to, you need to understand that a surprising amount of stuff will happen in less than 5 rounds of combat. The more players you have, the longer it will take to get through a round. Finally, you need to understand that it's not fun sitting and watching a combat unfold while you can do nothing. For this you need to factor in those actions that can happen DURING combat, and people trying to arrive in combat. Someone driving to a fight can sit and wait for 2 hours to get into a fight scene, only to when they ask find out that it's only been 30 seconds since they got the call. If they're lucky they may have found their keys and reached the door by then. What about the police arriving? If things like this are going to be common in the game you're making, you may want to make the timers longer. Otherwise, the GM (or your system) is going to have to find other ways to keep those people not directly involved int he fight entertained as they do their extended actions.
Round timers, as said, are important for controlling the flow of things. Matched to a right group size, game type, and GM they can really help combat with feeling alive and visceral as it was meant to be. Done wrong, it can feel arbitrary and confusing. When designing a game, and looking at how long a round should be, you need to keep in mind what should be happening in an average round. How big are the fights going to be? How many players is the game designed for? These are all important things. Finding the right balance, combined with what you want happening in the round is where the answer lays. Just always keep in mind that nothing is perfect when it comes to games, there is give and take for every decision made, so you need to find the one that feels right for your game. Even if that is to disregard round timers completely like games like Mutants and Masterminds does, where each players turn is the equivalent of 1 frame in a comic book.