If a zombie gets close enough to the player, it will start following them until it starts causing damage. This isn't too much of a problem since they walk fairly slow and they have 20 health points (10 hearts). When killed, they drop 1-2 rotten flesh and/or Rare Item's such as Potato, carrots. They also are one of the smartest mobs. However, like most mobs, if you are under them in a tunnel, they will just get on top of the tunnel and start jumping. They can jump about 2 blocks to get to you.
When a zombie comes in contact with light from the sun, they will catch on fire and take damage. It is a good idea to wait until morning to hunt for zombies because they will already be taking damage from the sun.
Besides spawning randomly, zombies will also spawn from mob spawners, making mob spawners a convenient way to get their drops.
Once you kill a zombie it will drop rotten flesh which is pretty much useless. You can eat it but it doesn't do much other then give you a hunger effect.