What are some of the reasons faucets drip when its turned off?

A bad washer if it is the type that uses them.

Like chuck says it can be a washer or a cartridge depends on the faucet, if you give us the brand and the model number of the faucet and tell us it is a kitchen, bath sink, single or 2 handle faucet, we will tell you what needs to be done, Thanks.

John

It could be a washer crackedstem or a bad seat depends on what you got

Pretty much all of the above or it could also be a scratch inside the valve body from when you changed or installed a new cartridge. Are you asking this because you did a repair and now it drips?