Conduction is the method of transferring heat within a body or from one body to another caused by the heat transfer by molecules vibrating at their mean positions. The bodies involved in the heat transfer must be in contact with one another. When heat is transferred from one location to another, there is no actual movement of matter.
Conduction is most frequent in solids, where molecules are kept together by intermolecular forces of attraction, causing them to only vibrate about their mean locations as they receive heat energy and hence pass it on to the surrounding molecules via vibrations.
Convection is a heat transport method that occurs mostly in liquids and gases. Heat transfer occurs in this mechanism when the substance is moved from one location within the body to another. Eg. It is often witnessed bubbles and currents arise in boiling water. (Fig 1)
This is an excellent illustration of the convection process. The hot water at the bottom gets lighter and rises, causing the colder, denser water at the top to descend and heat up.
Another method of heat transport is radiation. It works without the use of a medium and can even transfer heat in a vacuum. Electromagnetic waves are used in this way to transfer heat from one location to another. Radiation is how the sun's heat and light reach our planet in our solar system.
Radiation is, in fact, the most powerful means of heat transport. We feel warm without touching the burning wood when we sit near a fire in the winter. Only radiation makes this possible.