Also called diffusion, conduction is a type of heat transfer that occurs between two objects that are in direct contact with one another. An animation to illustrate conduction could be as basic as showing a pot bubbling on a stove. When the burner is lit below the pot, heat is transferred to the pot, which is made of a conductive material that evenly distributes the heat around to pot to effectively cook the pot's contents.
Convection is the process of heat transfer between an object and its environment through the movement of fluids. In the context of convection, a fluid is defined as any substance that loses shape under stress, which includes liquids and gases. A heat transfer animation for illustrating convection could be a fireplace or wood stove. Once lit or turned on, the heat that is generated rises up and circulates around the object of its origin. Clicking on the stove or fireplace animation to make it "turn on" would be one way of illustrating convection through animation.
Radiation is the type of heat transfer that occurs through electromagnetic waves. Heat is transferred when the electromagnetic waves come into contact with an object after traveling through space. An animation of the sun's warming effect on skin or structures is one way to illustrate radiation. The animation might animate the normally invisible waves that the sun emits as they travel through space until they reach a person or object, which is in turn warmed by coming into contact with the waves.
Mass transfer is a secondary type of heat transfer. It is the exchange of heat between two objects by physically moving the heat-emitting object to a new location. An animation depicting mass transfer could show an object that had been heated in a fire -- for example, a blacksmithing iron -- traveling from one work surface to another. The second work surface -- the endpoint of the iron's trip -- would become heated by coming into contact with the hot mass of the heated iron.