Your car may have plastic guards underneath to stop water from spraying up on sensitive components, but everything under the hood is typically waterproof to some degree. That said, the only "damage" I have ever seen is the spray of water loosening or tearing off plastic guards that are not properly fastened or half-way ripped off already.
The basic short answer is, it shouldn't break anything that isn't somewhat broken already. Cracked light housings could get wet, etc.
If the water is deep enough that there is a huge rush of water over the front of the car or the engine bay, you could suck some water into the intake, but then again, the intake/air filter housing is made to deal with this as well.
Added: The heat from the brake system should dry them very quickly. There will be a momentary loss of a little braking power, but shouldn't be very noticeable. Most of the splash will be directed away from the wheels, so the brakes might not even get wet.
I personally love to hit puddles at high speed to see the big spray. My wife hates it.
Edit based on comments...
For safety, don't speed on residential streets and never hit a big puddle unless you've driven over the same spot in dry weather. Chances for hydroplaning are great, so go over puddles in a straight line and avoid steering or braking input until the puddle is crossed.
Large waves of water can "hydrolock" the car if water is sucked into the engine, which is bad. The intake is made for splashes, not waterfalls.