The concept of designing D&D around a certain number of encounters per day first came to the fore in D&D 3.0. I think everyone reading this is probably familiar with the concept already, but I'd like to drill down on it a bit, because this approach completely changed D&D, and made it unlike any other roleplaying game I can name. I'll start by saying that I neither approve nor disapprove of this approach - I can enjoy a game that embraces this idea as much as I can one that has rejected it.
In 2e and earlier editions, the general idea of an adventure was that low-level spellcasters had very little spellcasting that they could do in a day. Fighters and rogues, on the other hand, could do their thing all day - as I've said in the past, "sword mana never runs out." At low levels, the casters need to work out another solution; at high levels, they'd be hard-pressed to run out of spells without purely wasting them. A big part of the caster's reward for gaining a level is increased staying power in encounters. At low levels, parties keep going with the adventure even when the wizard is out of spells. At later levels, it's a bit more tempting to stop what you're doing and hole up once the casters are out of spells; in particular, you've grown much more dependent on the cleric's healing to get you back to a safer number of hit points. Still, the designers had no concept of pacing the game around a certain number of encounters in a day; they expected players to play through the adventure, and gave DMs wandering monster tables to keep the PCs from resting whenever they liked. For want of terminology, I'll call this a story-paced game - its pacing is determined by the flow of the story at the table.
D&D 3.x is the strange middle-ground edition here: still using Vancian magic, with its per-day spells, but conscious of every spellcaster's desire to spend their combat actions casting spells, rather than holding back and conserving spells tightly. (An exception is made here for all the various types of casters who actually have significant physical combat ability, such as most clerics and druids.) One of the main reasons that the designers introduced 0th level spells was to give even the lowest-level casters a bit more staying power. I can only imagine that introducing bonus spells from high Intelligence for wizards was motivated by the same goal. I generally found that by around 5th level I had enough spells that I didn't need to look elsewhere for something to do during a combat round, unless we were pushing our fifth or sixth encounter in a day.
The concept of an "expected" number of encounters per day was used, in 3.x, as a very loose way to balance weapon-wielders and spellcasters in the early going. (To contrast it with story-paced games, I'll call this an action-paced game; that is, designed to work more like an action flick.) Weapon-wielders manage their hit point total, and that's about it; they essentially have the same amount of "fun" (that is, they are at their peak performance) no matter how many encounters there are in a day. Spellcasters can "go nova" (as the slang goes) if they don't think there will be more encounters in the day - essentially, they can perform above what we might otherwise regard as their optimum performance. This was especially true of psions and wilders. It wasn't until much later in WotC's development of 3.x that they did much with giving weapon-wielders any form of per-encounter or per-day abilities (here I'm thinking of the Tome of Battle), though they also experimented with various kinds of melee brutes who employed spellcasting as central parts of their combat style - psychic warriors, soulknives, and duskblades all come to mind.
The thing about a consciously action-paced game is that the phenomenon of the 15-minute adventuring day became widespread, and widely discussed. Maybe this was a major problem in other folks' 2e-and-prior D&D games, and maybe it wasn't; I don't particularly recall it as being a problem, but the last of those campaigns was, um, more than eleven years ago now, and my recollection may be spotty. Once the DM is particularly conscious of the idea that an adventuring day "should" contain about four encounters, it changes the way the DM tells stories. Yet if the DM ignores the encounters-per-day suggestions, other aspects of the 3.x's balance suffer. Specifically, the guidelines for Encounter Level go out the window, and the DM is pretty much on his own to figure out how much danger the PCs can handle in that day. This is one of those things that's obvious after the fact, but took me a painfully long time to figure out in running games: if you run fewer encounters, you can run 1-2 big encounters - just be very careful about how you increase the threat level of the enemy, because it's not that hard to just overwhelm the PCs. (Digression: be careful about that statement too, and remember that a big pile of enemies can sometimes just be vaporized by a single fireball. The best possible thing you can do for game balance when statting encounters is to just do a bunch of different things.)
After the cut: moving on to 4e, and how it changes everything; also, how encounters-per-day design doesn't factor into other games, from Mage to Dust to Dust.
So in non-Essentials 4e (hereafter 4e as opposed to 4E, because I'm tired of typing "non-Essentials 4e"), a character's powers can be at-will, encounter, or daily. Characters start off with one daily power, and gain additional daily powers quite rarely. As you'd expect, daily powers can be devastating in their effect, and even on a poor attack roll, daily powers always have some effect (or are Reliable, so that it's not expended on a miss). This setup puts fighters and wizards on equal footing, so that it's not always the casters who are starting to encourage the team to hole up so they can recover their powers.
The crucial per-day currency, though, isn't the daily powers - it's the healing surges. Healing surges are, I think, the game's primary focus of attrition. Basically, the DM can try to kill the players quickly, with enough damage to overwhelm their ability to heal (or kill them before they have a chance to heal); or he can try to grind them down by forcing them to spend healing surges and preventing them from resting. All characters benefit about equally from days with fewer-than-normal encounters and suffer about equally from more encounters than normal.
As I've said, the game's balance comes to revolve around this idea; the only reason 2e's balance didn't was that the game was fundamentally not balanced around combat at all. If the DM starts thinking about how balance revolves around combat, it significantly changes how he runs the game: it becomes a better action game, I think, but verisimilitude becomes harder to maintain. See, there's not much point in a single wandering encounter out in the wilds, unless it's overwhelmingly dangerous, and I found that I wanted every encounter to have a point, because it's likely to be a third to a half of evening of play. Likewise ambushes: a party that has not faced any other encounters in the day is remarkably difficult to threaten.
I found that my D&D games took on a more structured flow, as I tried to figure out the three or four encounters I'd need to throw in the players' way before they faced the climactic encounter of the adventure. I need to put forth considerable effort to keep them on that track, too - while I don't especially care if they skip an encounter, they might be fucked if they run into two encounters at once through some accident of exploration. There's a lot to be said for letting the dice fall where they may and letting emergent storytelling happen (I do read OSR blogs), but there's also the fact that DMing is about restraint - the DM can always hit harder.
By way of contrast, there's my Mage: the Awakening chronicle, most definitely a story-paced game. As I discussed in this post, Mage doesn't bother with any concept of encounter balance, and just assumes the Storyteller knows not to send psychopathic Scelesti archmages after newly-Awakened PCs. The PCs start off able to hold 10 mana, and increase from there as they gain Gnosis; since they spent a total of three Merit dots on their Hallow and their Hallow can store up to fourteen days' production worth of Tass, it's a lot like their Mana pools are per-day resources. Of course, so many of their spells are zero-cost that Especially recently, they've kept busy enough that their Willpower pools aren't always full at the start of a day of something interesting happening, but I think they're keenly aware that events in the setting are advancing on a timeline. If they have just one encounter in a day that causes them to spend any resources, though, it's not really a big deal.
This lets me worry less about how I'm going to include the right challenge level of encounters in a day and focus more on the natural flow of the story. As a result, Mage isn't a great action game - that is to say, it's hard to present compelling tactical challenges. This has a lot to do with how the game doesn't discuss scaling, encounter balance, per-day capability, or the like. It does just fine at presenting compelling strategic challenges and puzzles, though, in a more naturalistic way than D&D does.
LARP game balance is still another matter. Other than maybe IFGS (and only maybe in that case), I've never heard of a LARP spending any serious mental energy on designing around a particular number of encounters in a day. In most boffer LARPs, you have a relatively high degree of control over how many encounters you participate in over the course of a day, and players who prefer to avoid combat encounters might well do so without ruining their event. In DtD, nonmagical abilities are almost always at-will or per-encounter, resetting with ten minutes of non-combat activity, while spellcasting is per-day, and consumable items are... aptly named. This is why casters of all kinds are so strongly advised to have some kind of weaponry if they intend to participate in every combat that crosses their paths: even a celestial of the Fifth Circle, with a daily pool of 80 mana, can cast his strongest spell "only" sixteen times. This is not generally seen as the wisest use of mana, but it's certainly impressive; faced with opponents who have any kind of relevant defenses, it's a great way to run out of mana well in advance of the action of the event.
This post is long enough that I don't want to go into it in detail, but I've played LARPs that chose per-encounter design for all abilities, so as to create an action-paced game. Some early design discussions of DtD toyed with the idea of moving in that direction as well, but we eventually rejected it, taking into account the difficulty of flinging monsters at PCs quickly enough to cause them to suffer any attrition if the PCs reset more abilities, or reset them more quickly.
Naturally, I'll be interested to see what D&D 5e does with per-encounter and per-day design. Given their stated goal of drawing OSR players back into the fold, I would expect to see a significant return to Vancian daily effects. I wouldn't expect the survival of many of 4e's pacing innovations: healing surges, action points, and limits to the number of daily effects a single character can use from magic items. (The last of these has been the point of much back-and-forth in the last year or so of 4e.)