The lights are on
When I looked for help with this boss battle and everyone online claimed it was luck; NOT TRUE.
Here's how you beat the Reaper:
Round 1) As he's shooting strafe left; regain your ground towards the left when he's not; rinse and repeat
Round 2) This time strafe the other way each time he's firing; regain ground to the right when he's not; repeat
Round 3) (Little bit harder) This time he alternates firing paths STRAFE RIGHT THE FIRST TIME; then left; then right; you should know what alternating means; (I didn't find myself having to regain ground nearly as often during this round)
Round 4) The only way to beat this round is if you are targeting the center of his eye, in which case should take like 3 seconds
EDIT: In my experience, I always stayed to the left, waited for him to fire, and kept dodging right until he stopped. Then I would aim at him, and if I couldn't hit him in one, then I would run back before he fired again. I finished easily enough. Hope this helps. -Rekoin
EDIT: Contrary to the original poster, the Reaper only fires in the direction that Shepard is currently at; there's no predetermined pattern to this other than that. Simply dodge it and run for a couple seconds, then aim at its laser until it prepares to fire its laser again- usually time it so that you stop just as it's almost finished firing its laser (it should be too high to hit you). Make sure you never let it corner you.
You won't be able to charge completely at once, only gradually, (especially on higher difficulties) so this will take quite a few turns each time. After hitting the Reaper with the laser, it will advance one step forward and you'll have to repeat the process until it's directly in front of you. You'll naturally have to aim at it - quickly - until the laser activates and destroys it once and for all. -Orochi