Annihilation is a game mode in which to win one must destroy every unit that belongs to his enemy. This option is set before a skirmish game.
This game mode is played more by newer players in case an "accident" happens to their ACU. 95% of online matches are assassination and you should get into the habit of playing assassination if you want to climb your way up the ladder. It is much more easy than assassination because you don't have an ACU to protect. It also gives rise to tactics that assassination players do not usually have to contend with. For example, overall unit efficacy becomes more important instead of just a unit's alpha strike capability versus the enemy ACU. It also makes your ACU's death explosion a viable threat, especially if you can sneak it within the confines of your opponent's base when it dies.
Annihilation game mode is favored by those players who want a traditional Real Time Strategy game experience instead of being encumbered by the Real Time Ladder Climb mechanic.