Variable Player Goals

Sometimes, not all the players are going for the same thing. There are essentially two forms of this.

In the milder form, each player is going for the same sort of thing, but what precisely they require varies. For instance, each player may be trying to get two treasures, but which two treasures is different for each player.

In the more extreme form, each player is going for something different entirely. One player is trying to accumulate as much money as possible, while another is trying to eliminate the player to his left. These are often accompanied with the player having varying special abilities that help to accomplish these specific goals.

In either case, these are typically assigned randomly at the beginning of the game. Sometimes, each player's objective is secret, though this is hard to pull off if they are supposed to be matched up with the player's specific abilities. There is also often a generic win condition that any player can accomplish.

A common twist on the more extreme form is that one player has no special win condition; instead, his generic win condition is made easier. A less common twist is that one player wins automatically if the game would go to final scoring or the game's equivalent.

Examples of Variable Player Goals include:

Variable but formulaic win conditions

Chrononauts - Each player can win by accomplishing their ID, which requires getting certain events on the timeline, their mission, which requires getting specific artifacts, or by getting 10 cards in hand.
The Amazeing Labyrinth (and related games) - each player is going for different treasures.
In Legend of the Five Rings, there are four possible ways to win the CCG. You can either go military victory, and destroy all enemy provinces, Honor victory, by getting your Honor above 40, Dishonor victory, by getting your opponent's honor at -19, and Enlightenment victory, by playing all Five Rings onto the field.
The Civilization games have military victory (destroy all other civilizations), culture victory (get your Culture high enough to create a utopia), Technological victory (build a spacecraft to send your civilization to the stars), and a diplomatic victory (convince all your opponents to make you president of the world). No civilization is barred from achieving any win condition, but all have traits that make pursuing certain goals easier than others.

Highly variable win conditions

Secret Mission Risk.
Dune - Not every side gets a special victory condition, but includes two sides (Spacing Guild and Fremen) which win in case of final scoring; the Fremen require additional conditions to be met, however, and the Spacing Guild wins only if they are not (or if the Fremen are not playing). Also includes the Bene Gesserit, who can steal the win if they correctly guess before the game starts who will win the game and on what turn.
Illuminati - Runs the gamut from destruction (Servants of Cthulu), accumulating raw power (Bavarians), getting a crapton of money (Gnomes of Zurich), collecting one of every alignment (Discordia), or a choose-your-own secret goal (UFOs). Also includes Church Of The Subgenius as the side who have an easier time with the ordinary condition.
Twilight Imperium.
StarCraft: The Board Game - The Aldaris faction combines both common twists. However, instead of his own score required for ordinary victory being lower, if the Aldaris faction is playing, everyone else's score required for ordinary victory is higher.
The victory condition of Avalon Hill's Magic Realm was labryinthine, but essentially came down to getting a sufficiently high total among the game's several scoring tracks, some of them mutually exclusive. Certain characters were significantly better at raising certain variables, thus giving different tracks to victory.
The card game Bang! randomly assigns roles to each player. Each role has a specific victory condition that boils down to killing someone else, but since the Sheriff is the only role revealed at the start of the game, part of the challenge is figuring out which players are your intended targets.
Final Fantasy Crystal Chronicles uses this to add a competitive element to a primarily co-operative game- everyone has to work together to complete the level, but each player has a unique (and secret) secondary objective. Bonuses are awarded to whoever did the best job of completing their hidden mission, which will often involve hindering the team somewhat.
- However, it may also be in the team's interest to tell each other their objectives, because the higher the team's combined score is, the better rewards available at the end of the level will be. So it is in the team's best interest to compromise to get the best rewards.
In one of the Star Wars novels, Luke has developed one of these games for his Jedi students. He explains that it's actually a training exercise in diplomacy for the referee, who has to figure out both teams' victory conditions and, assuming they're not mutually exclusive, try to help everyone win.
In the upcoming Ankh-Morpork board game, three of the characters (Lords Selachii, Rust and de Worde) have a standard "control X areas" goal, Lord Vetinari must place X minions on the board, Dragon wins by driving the city into chaos, Chrysoprase wins by collecting enough money, and Sam Vimes wins by stopping anyone else from winning within the time limit. Of course, the players don't know which roles are in the game, let alone who holds which role.
In the Battlestar Galactica board game, the humans win by travelling a certain distance without running out of resources, suffering too much damage, or being boarded by Centurions. The hidden Cylons win by preventing this. On top of this, one expansion adds personal goals for humans, encouraging them to take actions (often ones which hinder the team) in order to avoid penalties.
The board game Shadow Hunters is built around this. Hunters want to kill Shadows, and vice versa, but every game will also feature neutrals, whose goals include "survive to the end of the game", "be the first to die", "gather four powerful artifacts", "kill a powerful Shadow or a powerful Hunter", and others.
In the Mario Party series, most 1 vs. 3 minigames have a different objective for the single player than for the 3 players, for the reason that it would be unfair otherwise.
In the Global Conquest mode for Command and Conquer 3: Kane's Wrath, every faction has its own victory condition: GDI wins by having their territory cover a certain percentage of the land area; Nod wins by bringing a certain number of cities to complete unrest; and the Scrin win by building a certain number of strategic structures. There's also the universal win condition of simply eliminating the other factions in combat.