Bribery: "The offering, giving, receiving, or soliciting of something of value for the purpose of influencing the action of an official in the discharge of his or her public or legal duties."If you want a public or legal official to do something illegal or lie for you, you can offer them money or pay them to do it.Extortion: "The obtaining of property from another induced by wrongful use of actual or threatened force, violence, or fear, or under color of official right."Extortion then is taking property (usually money) from someone by threatening them or pretending it's your legal right (as a government official, police officer, etc).Blackmail: "The crime involving a threat for purposes of compelling a person to do an act against his or her will, or for purposes of taking the person's money or property."And blackmail is when you threaten someone enough for them to feel they have to pay you or do something in order for you not to carry out the threat.
So, they all involve some sort of deception or threat in order for a criminal to get what s/he wants from another person, they just all accomplish that in their own ways.