When someone says, "I have a cold," what he or she means is, "There is something in my body that is causing me to have the set of symptoms that we call a 'cold.'" The set of symptoms normally includes things like a runny nose, sneezing, coughing, "chills" and a headache. It does not include a fever -- normally, if there is a fever it's called "the flu."
There are many different viruses that can cause cold symptoms, but about half of the time a cold is caused by a class of viruses called rhinoviruses.
The rhinovirus gets into the cells lining your nose and starts reproducing. It arrives from other people -- it is not cold weather that causes a cold, but the fact that cold weather causes people to congregate together indoors, which makes transmission of the virus easier. The virus generally moves from someone else's hands to your hands (either directly or through some intermediate surface like a door knob), and from your hands into your nose or eyes.
Your body reacts to the presence of the virus with its immune system. The article How Your Immune System Works talks about infectious diseases and how your immune system deals with them. In the case of a cold, the immune system opens up blood vessels through inflammation and also increases mucus secretions. These two processes give you the runny nose and the stuffy feeling. The irritation caused by the virus and all of the fluid causes sneezing. If the virus makes it into the cells lining the lungs, then they start producing fluid and mucus as well, which produces the cough.
As the immune system gears up over several days and fights the virus, the mucus thickens and changes color with dead cells (a form of pus, really). Eventually, the immune system eliminates the virus completely and you are well again!
For more information, see the links below.