Required goals must be accomplished to finish (or win) a game. There are no workarounds or alternatives for these goals; they are frequently referred to as bottle-necks that impede a players progress in a game. Required goals can be considered as those goals that are on the critical-path towards success with the game.
The existence of required goals is not necessarily associated with linearity in games. In other words, it is possible for a game to have a set of required goals that must all be accomplished though the order in which they must be tackled is left to the players discretion.
Super Mario Bros
Super Mario Bros provides a strong example of required goals. The goal in the majority of the levels is simply to get to the end. The other ones require that you defeat a boss. In both cases the only way to proceed is to complete these goals.
in Goldeneye 64, each level has its own set of mission objectives, which include thing like picking up a certain item or destroying a certain object. Unless every single one of these goals are met, reaching the end of a level will simply result in the player having to start that level over.
Kirby's Adventure is a prime example of the idea of "bottle-neck" goals for game progression. Kirby is only allowed to enter one door (to a level) at a time and is only allowed to progress to the next door when the first is beaten.
In Diablo 2, play cannot progress past the current act until the player defeats the Boss for that act. Within acts there is some freedom (barely) of which task to complete first, but for the most part each of these must also be completed in sequence. Once a player defeats the boss for the given act, they can move on to the next act but this can sometimes be frustrating as players are forced to increase their character's skill in order to defeat the boss (confined to the areas leading up to the boss's location).
Splinter Cell:Double Agent
In Splinter Cell: Double Agent, each mission has its own set of primary objectives which you must complete in order to gain trust with various factions, otherwise you will fail the game. The more objectives you complete with one faction, the more you lose trust with another.
Need for Speed
Need For Speed also provides a strong example of required goals. Winning the race is usually the goal and its the only way. However, there are some variations like winning a certain number of the races in the main campaign, so it isn't necesarry to win ALL the races, just a certain amount.
Black & White
In Black & White, there are certain activities that the player is supposed to complete in order to progress in the game (for example, visit Nemesis' gargantuan ex-Creature wandering about the first island to learn to fight and such). If the player desires, however, they can forego this and simply attend to the original village's needs without progressing the game's storyline. This leaves the player in something of a state of stasis, but a player can technically consider themselves successful at the game by simply keeping that first village happy and healthy without progressing any further in the overall story.