The League of Legends Devs recently posted a design blog about what went into developing the quests, and why.
One of the major objectives of the quest system was to prevent the
emergence of a metagame that promoted capturing three command points
and turtling to hold them for the remainder of the battle. Turtling
strategies like this one are contrary to the high-energy combat that
we wanted to promote with the new game mode. So, in order to prevent
these sorts of stratagems from emerging in League of Legends:
Dominion, we instituted quests to allow us to alter the strategic
value of each command point on the fly.
A quest triggers at random intervals (I've seen two in one match, before) and will always involve two adjacent capture points, one controlled by your team, and one controlled by the enemy.
Completing a quest, that is, capturing your target point before your enemies capture their target point, gives you a short duration 10% damage buff, and immediately deals 20 damage to the enemy nexus.