What should I do about faulty concrete work?

First off,  do NOT pay him any more money.   If he is not done and there already are problems you need to point out the problems IN WRITING so he can not come back and say you did not tell him you saw a problem.  Having water collect in a puddle is not normal,  there should be a slight slope so the water drains either down the driveway to the street or off to the side.      How he fixes it is up to him.   IMHO the best way would be for him to cut out 6 inches of the concrete right next to the garage door and put in a curtain drain along the entire front of the garage with it draining to the side.  

a grinding machine can be brought in to lower the edge where the concrete should be draining off.  Then, it might [unsure about your climate/exterior conditions] be possible to build up the edge next to the garage.  And, this possibility should have been addressed when the design was drawn up ... error on his part, imho [since he's the expert and you aren't].  {It might even be possible to build a slight lip there from some material such as epoxy and have it last ... sorry, i'm not a contractor and don't know these materials well.}

Getting him to do this may require the threat of an attorney ... so find one in the area who specializes in construction related problems ... sometimes just a stiff note from an attorney, coupled with his advice to you to withhold payment, is all that's required to get a contractor to do the job right.

--Grampa

Read the contract again.  Determine your "rights" for a repair.

First, check the "scope of work" in the original quote and the subsequent contract.  Proper slope and drainage should be in the specs.  This is very important.

Then, take a photo of the water not running off properly.  Let him know, and tell him he needs to fix it.  Make sure you get his acknowledgement in writing.  Since drainage is part of the job, it isn't properly completed without it. 

Since you haven't submitted your final payment yet, you have more leverage than if it hadn't been discovered until afterward.

The biggest problem most homeowners have with faulty work from contractors is when they've already paid before they discover the issue.