After I flush one of my toilets and the tank is full there is a constant dripping sound coming from the tank.Which part needs to be replaced?The flap, the ball, or the plastic part that is screwed to the base of the tank?

This sounds like a bad flapper. It could be the flapper itself or the lip it seals on. Raise the flapper and feel the lip with your finger tip, any scratches or ridges?

Okay, when I came home from work the toilet was constantly running.I raised the flap and did not feel any rough edges or cuts.Small amount of mildew but I cleaed it with a warm towel.When the toilet filled back up it continued to run.I pressed down on the flap and the water did not run anymore.Is this a sign the flap is bad?Should I remove the flap and go to Home Depot to get a replacement?Basically, are all flaps the same?

Yes, replace the flapper but they are not all the same, just close. Take the old one with you and look for one just like it. Better yet take the brand name of the toilet or the fill valve. The store should be able to match it

On top of Bob's instruction I would suggest that when you change the flapper make sure that the chain has 1/4" to 1/2" of slack see the image below, Thanks.

John

I say it's the fill valve. You don't hear a leaking flapper. You only hear the fill valve coming on to replace the water that has leaked out.
The dripping sound would come when the fill valve does not completely stop the flow and water overflows the overflow tube. The sound comes when the last few drops fall down the tube.

You said "toilet was constantly running", definitely fill valve.

When toilet constantly running, it's the fill valve, when toilet intermittently come on and shuts off it's the flapper.

Raid the piggy bank and replace both. Not matter which one you replace today, tomorrow the other will go bad.

Let's check to see just what part needs to be replaced.
After the toilet's set for a hour pull the lid and check the tank level. If it's spilling over into the white overflow tube then you have a bad ballcock.
If that's OK then tonight before bedtime put some food coloring in the tank. Next morning check the bowl. If it has color in it then one of two things.
1) Run your hand around the flat part of the flapper. If you have wavy edges replace the flapper.
2) If the flapper's OK then run you finger around the flappers seat, If you feel rough or nicked on it the flush valve's bad and needs replacing. Let me what you find. Good luck, Tom