Bird strikes happen all the time. Mostly they leave a smear on the paintwork, and very occasionally a dent. (Actually one dented the aluminium bonnet of my Land Rover, does that count)?
Most birds are small, and the engine simply eats them. But, as they get bigger, and if they hit the hot section and not the cold, then they can cause engine damage. Again, in most cases, the engine keeps on running, but if big enough, or in sufficient numbers, then they can kill the engine. The A320 in NYC showed the end result. Engines aren't the only places to be hit too...the RAAF lost an F-111 after a pelican strike on the canopy.
I've hit many a bird, but only once has it been an issue. On that occasion we were departing Perth for Singapore in a 767, when we ran into a flock of birds. The event recorder got it at 1,212 feet. I saw them go down the right hand side, and literally as I said 'birds', the engine note changed. Looking inside, both engines were running, but the right had reasonably high vibration. We pulled it back to about half power, cleaned up, and climbed to about 5,000 feet. Had a think about the state of the world, and came back and landed. We left the engine running, but at minimum power (just in case the right had some hits too). On the ground the engineers counted 12 hits. One of the flap fairings was missing entirely (obviously hit on the front, and then ripped off by air loads). The engine had lost two fan stator blades, and had four fan blades bent. Plus there was a hole in the engine casing. As it turned out, nothing had gone through the hot section of the engine, and it took the guys just 4 hours to fix it and have it run, ready for departure again.