Place a heated object in direct contact with the non-heated object. This is called conduction, and the non-heated object will eventually heat up. This is because the fast-moving molecules of the heated object will collide with the slow-moving molecules of the cool object, transferring some energy in the process. Eventually, most of (but not all) of the energy will be transferred and the cooler object will be hot.
Place a warm liquid or gas in a cooler liquid or gas. Since heat rises, the moving molecules from the heated substance will rise and collide with the cooler molecules, transferring energy to some and forcing others down.
Step outside on a warm day. The sun's warmth is transferred by radiation--the fast-moving atoms from the sun traveling at the speed of light and speeding up the electrons in the atoms on the planet. This is not necessarily molecular heat transfer, but it does follow the same principle, just applied to atoms instead.