Heat is kinetic energy on a molecular scale – that is the vibration of individual atoms and molecules. So when we think about something having a lot of heat, we mean that the atoms and molecules are vibrating a lot, and something with very little heat is made of molecules and atoms that are not vibrating much at all.
When two molecules or atoms are placed next to each other, they bump into each other as they vibrate. When they bump they can exchange some or all of their energy. That means that when a fast moving molecule bumps into a slow molecule it can give some of its energy to the slow one. The slow one will speed up and the fast one will slow down. That means that the slow one has picked up heat (and is now hotter) and the fast one has lost heat (and is now cooler).
This exchange of energy happens all the time and is the reason that heat travels through things. For instance, this is why putting a fire under the middle of a pot will heat up the entire pot. It is also the reason that putting something hot up against something cold will warm up the cold thing and cool down the hot one. For instance, if you put a cold spoon in hot cocoa, the spoon will heat up and the cocoa will cool down.
This transfer of heat happens until the temperatures are the same – that is, until the molecular vibrations match.
It is also true that the greater the difference in the speeds, the more quickly heat will transfer.
It is worth noting here that humans are very good at noticing heat transfer but really bad at determining temperature. Your nerves are NOT capable of determining how hot or cold something is, only whether it is hotter or colder than you are. An example of this is described here.