In dialogue, you can choose to Bribe someone. The Bribe option displays the amount of gold needed to successfully bribe the other character; If you can pay the price, the bribe succeeds.

The price of a bribe is based on your level and Speech Skill. At higher levels, bribes become increasingly expensive, although a high speech skill can offset this

Bribing, when present in a dialog, is a good way to get information, persuade an NPC to do something for you, or complete an objective, without having to resort to violence. Unlike other speech options such as Intimidate or Persuade, Bribing ALWAYS succeeds if you have the gold. So you should take advantage of this option whenever this is present.

Some quests can be completed faster if you bribe the right people.

Also you may want to look at the Bribery perk in the speech skill tree. The Bribery perk lets you bribe guards to ignore your crimes.