Break through the driveway's concrete with a jackhammer to get to the water underneath. Wear goggles and safety gear to help prevent injury from debris. You'll need to go only a few inches deep before it breaks through; be careful not to hit any piping if you have some underneath the driveway. If there is a location on the driveway where water collects, it is likely water is underneath as well because of the incline in that area, and the water pools need removal to prevent driveway damage. Likewise, places in the driveway that consistently crack likely have water under them.
Vacuum up the water using a shop vac. Water freezes in the winter and then expands so pools of water under the driveway cause the surface to crack over time. Preventing the water from getting under the driveway is important, but the shop vac removes puddles that are already there.
Refill the driveway hole with concrete mixture. Follow the package instructions when mixing the concrete with water to give you the most durable mixture possible. Smooth the mix with a trowel after you've filled the hole so it is even with the rest of the driveway. It will dry in several hours but fully hardens over the next week.