Optional goals are are communicated directly to the player, but are not necessary to the successful completion of the game. This of course, requires that the game have some sort of ending condition. Optional goals are understood to be optional and pursuable at the players discretion. It is important to note however, that at times, it may not be entirely clear to the player which goals are optional. In hindsight, he may be able to realize that a certain achieved goal was optional, but this may not always be the case.
At times it is hard to distinguish between required and optional goals when the difficulty of achieving a given required goal is dependent on completing certain optional goals. For example, in a typical RPG game it may be necessary for the player to complete a few Side-Quests in order for the characters he controls to improve their abilities enough to take on a critical-path goal. Thus, at times, optional goals may become critical depending on factors such as the players skill level.
Final Fantasy VI
In Final Fantasy VI, the game splits from the World of Balance to the World of Ruin partway through the game. In the World of Ruin, the player "re-starts" with only one of their characters. As the story progresses, the player will come across other characters and get them to re-join their party. However, the vast majority of these characters are actually optional from that point on. Theoretically speaking, as long as the player gets the airship and enough characters to form the three parties required for Kefka's Tower, they could finish the game without re-collecting a lot of their former party members. Getting all of the possible members for your party in the World of Ruin is an optional goal.
Additionally, like with most RPGs, FFVI has optional sidequests the player can embark on to gain items such as weapons, armor, relics, and Espers, as well as viewing scenes otherwise not present in the normal storyline.
Splinter Cell: Double Agent
In Splinter Cell: Double Agent, there are sets of objectives that are labeled as secondary, or optional in this sense. These objectives can be completed for added bonuses such as new gadgets or more trust with a faction, but they are not necessary for mission completion.
Street Fighter III: Third Strike
While playing SF III in single-player mode, you begin to collect points based on your performance within each round. More points area accumulated if the player performs any number of optional things within each round. KO'ing an a opponent with a Special Move, or without losing any health yourself, will garner more points. These extra points and, in fact, the entire point system are completely secordary to the main storyline and to the system by which you unlock new characters and extras. You unlock the extras merely by completing the game; the score and grade are irrelevant.
Command & Conquer: Tiberian Sun
In Command & Conquer: Tiberian Sun you have the option of choosing missions that will add support to the next primary mission. For example the last level of the Nod campaign you can choose to directly play the campaign or you can play a mission to steal the ion cannon codes. Once finished you will get the ion cannon on the next mission. We consider this a weak example due to the role that completing the optional mission has on the final level (makes it easier).