Conduction occurs when two things that have different temperatures come into contact with each other. It occurs because the heat in the hotter object transfers into the material of the object that is not as hot.
The easiest example to understand heat conduction is the way that water gets hot on a stove. The stove starts out by being extremely hot. It heats to a high temperature quickly. Water is placed in a pot and the pot is placed on the stove. The heat from the stove transfers to the pot and the pot begins to get hot quickly, depending on its original temperature. The water that is in the pot will eventually get hotter as the heat is transferred from the stove to the pot and into the mass of water that sits within the pot.
Conduction can also occur when a temperature is cold. When something that is temperature conductive is at an extremely cold temperature, it can make the things it touches even colder. One way to think of this is the winter weather. When a person is standing outside in the winter, it generally does not feel as cold as the true temperature. If the person were to pick up a piece of snow or even some cold ground, their hands would get colder much more quickly. The cold from the ground they picked up is transferred onto their body and can make them colder.