[Legolas]

Combat is broken up into three attack moments. If you have one apr, you attack in the first moment of the round, then miss two. If you have two apr, you attack in the first and second moment and miss the third. With three apr, you cover all attack moments. With four, you attack twice in the first moment, and once in the other two. And so on.
Your main hand attacks are always resolved first, your off-hand attacks second. So suppose you were fighting with a longsword and shortsword, had 4 attacks per round and another 2 from improved two weapon fighting, and were hasted for an extra attack per round, you'd strike three times with your main hand, then twice with main hand and then twice with off-hand.

The initiative roll determines at what exact point of the round your fight moments occur, so you might strike almost simultaneously or take turns depending on the occasion.
Usually, in the example above, A strikes twice, then B once, then A once, then B once, then A once, B has no attacks left, and then it starts all over again.