You pay a bribe to get someone to do something they wouldn't do otherwise. It's usually dishonest and often criminal.
Bribes can be utterly corrupt and illegal — like when you offer a bribe to a politician so she'll make a decision in your favor. On the other hand, a bribe can be somewhat benign — like when your mom bribes you with an ice cream if you play with your little brother. When you bribe someone, you can say you're "greasing their palm."