Not to belabor the obvious, but Victory Conditions are what make a game a game. They constrict the game's scope and dictate its structure. Victory Conditions set goals for the players, and their strategies and tactics are generally crafted with those ends in mind. Probably the single most important thing a designer does (besides looking stunningly handsome) is decide how the game ends, and how the game is won.
Sometimes these are two different things that exist somewhat independently of one another. The game might end, for example, after you've played through a certain number of Game Turns. This is pretty common, especially for games that used a fixed time scale: each turn is a day, each turn is two hours, each turn is fifteen-point-four-six minutes. The battle lasted for X amount of time, and each turn is one-twelfth of X, ergo the game is twelve turns long. The game ends at the conclusion of the twelfth turn, and at that point you tally Victory Points.
Maybe only one side scores VP, or the successes of his opponent result in negative VP, or both players earn VP separately and the difference is used to determine victory. Whatever the case, it's common with this model to have "levels of victory", ranging from tactical to decisive, or pyrrhic to overwhelming, or whatever flavor the designer fancies.
What's really keen about this sort of approach is that it allows you to compensate for advantages enjoyed by one side or the other. If, for example, the Confederate Player is better able to score VP than his Union counterpart, you don't want the game to go to "whoever has the most points", because the Confederate Player is going to have the most points in every game. Whereas if the boys in butternut have to score 10 more VPs than the Union to qualify for a decisive victory and 6 for a tactical, and anything 5 or below is a Union Tactical Victory, it's not guaranteed that the Confederate Player is going to win every time, and the pressure is on the Confederate Player not only to do well, but to replicate the historical result.
This method can also be used without adopting fixed scale turns. In fact, most of my own designs don't use a fixed number of turns, or a fixed time scale, at all. Instead, the game keeps going until one of the players has hit a certain number of VP, at which point they win. While this doesn't allow for levels of victory in the tactical-to-decisive tradition - since as soon as they cross the first threshold, it's going to end the game - it does allow for different VP targets for each player, putting an onus on one side or the other to recreate or best the historical result.
In the case of my Shields & Swords II designs, these VP targets are often two-tiered. For example, in the upcoming Battles on the Ice, the Novgorod Player wins the game if (1) he has scored 25 VP, and 5 more than the Livonian Player, or (2) he has scored 30 VP. The Livonian Player, on the other hand, needs to score 25 VP (and 5 more than the Novgorod), or he needs to eliminate all eight of the Novgorod Player's elite Druzhina. Once one of the players meets their victory condition, the game ends and they win it.
Of course, this brings up the question, what happens if both players achieve one of their victory conditions at the same time? Well, that's the thing I really like, and am probably inordinately proud of, about the S&S II series: each player checks for victory at the end of his opponent's turn. This means that if your opponent has hit that "25, and 5 more" threshold by the end of his turn, you have the whole of your turn to close the gap, and if you do, he has the whole of his next turn to try to open it up again. In my experience, this has made the last few turns fairly competitive and tense provided the players are of equal skill.
It also allows me to time the battles. A game that goes to 25 VP feels a lot different than one that goes to 30 or 35. The VP thresholds for many of the S&S II games are usually fairly close, actually, even for historically lopsided battles like Grunwald. There's a reason for this, specific to medieval and ancient warfare; the vast majority of the blood-letting is done after the battle has been decided, while the losing side is trying desperately to get away. And so the S&S II series only follows the action up to the tipping point, with the wanton butchering, sacking, and taking of captives happening long after the players have put the counters back in the box.
Of course, some games don't use VP at all, instead having victory determined by control of this hex or that one. This can work in certain cases, and I'm a fan of using it as an alternative victory condition so as to model historical imperatives. For example, in both The Grunwald Swords and the S&S I title Our Royal Bones, there are hexes in the enemy's rear that will give one side an automatic victory regardless of score. I don't actually see those automatic victories all that often, simply because the threat of it is enough to make the other player super-paranoid about protecting his rear at all costs. And I used something similar in Blood in the Fog, in that control of a specific hex (the Sandbag Battery) moves the Russians one step closer to victory at the end of every turn.
But if that's the only victory condition - control this hex to win, and if you don't, you lose - I'm actually not such a big fan of that myself. Part of it is that it feels kind of gamey. One of my complaints with the old GDW Battle of the Alma game is that victory hinged entirely on a couple of road hexes. There would be games I played where at the end of the last Game Turn, one or two lone British units had taken the hex, winning the game despite being surrounded by the Russian army. "Oh no, those guys are on the road, I guess we better give up now." Granted, that could easily be fixed by requiring that some kind of line of communication extend from the victory hex to the British end of the map. But as a general rule, I'm not a huge fan of "grab that hex at all costs", though there are some exceptions: I dig it in Plan 1919, for example, and while there aren't any hexes, I think it works in Supply Lines of the American Revolution as a decent approximation of British strategic aims.
Nor am I really enamored with games where the victory conditions for one side amount to "Stop the other guy from winning", though I think this is more of a personal preference born out of my own work as a designer. For one thing, this kind of approach is of course completely incompatible with an unfixed number of Game Turns like the ones I tend to employ. For another, a lot of my games are inordinately obsessed with tempo, one player's ability to force the other to react to what he's doing, and the other player's ability to turn the tables. If a player has no strategic aims other than stopping the other guy, there's nothing for him to pursue and no reason for him to stop reacting and seize the tempo. Such a game encourages a player to adopt the Brown Ben Plumm strategy: he does not play to win; he plays so as not to lose.
For me, that's not very dynamic. That doesn't mean that a game that uses that kind of victory condition for a defending side can't be enjoyable; I've played many games that use some variant of it that are absolutely aces. (Which, Mary?, Absolutely Aces sounds like a great title for an aviation game. Put this on my to-do list, please.) Several of the Horse & Musket scenarios use some version of this, and I've found them all to be enjoyable and replayable experiences. Frank Chadwick's Beda Fomm, which I recently acquired, has a really rather clever innovation on this old chestnut, in that the side that wins "by stopping the other guy" is the side that's doing almost all of the attacking. It's only the poor, beleaguered Italians that earn VP in that one.
Of course, there are lots of other ways to approach victory conditions, each with their own pluses and minuses. I read a week or two back about someone who posited the unusual idea of a game with no victory conditions at all. IIRC, the argument ran something like, most players never finish the game anyway, and will just concede when it's clear the other guy is going to win. I thought that sounded interesting but also kind of nuts, in that Robert Frost tennis-without-a-net kind of way. As the venerable Dean Essig wrote in his rules for Last Chance for Victory, "For most players, telling them that they will recognize it when they win is plenty - but a minority will take that as a license to do something pointless". Well-crafted victory conditions are one of the best tools a designer has to focus his or her design on what they think was important or noteworthy about the conflict, and I can't see a game really working if someone abrogates such a key responsibility.