How does the body fight off an infection? Like when someone catches a cold.

The body has many ways in which it deals with 'foreign invaders', like a rhinovirus (the virus responsible for the common cold). Once an infective micro-organism enters the body, lots of different cell types, known as 'immune cells' are activated so that they can help clear the infection. There are two main 'arms' to the immune response; the innate response, and the adaptive response, which carry out different functions.
In the innate immune response, lots of cells (such as macrophages and Natural T Killer Cells) are activated which directly attack and kill anything foreign in sight, such as the invading micro-organisms. When activated, these innate destructive cells also become able to activate the cells of the 'adaptive' immune response. The activation of this adaptive immune response leads to highly efficient and specific clearance of the infection from the body; this process involved many different cell types, and the production of immune proteins called 'Antibodies' which lead to destruction of the infective microorganisms.
The action of the immune response through innate and adaptive immunity leads to effective clearance of invading microorganisms, and the infected person recovers from infection.