Influenza virus is what causes flu or common cold and it is called a contagious disease because of the viruses's ability to reproduce faster once in the host body and it's ability to spread through air,spit, sneezing,cough etc. These also acts as the contagion mechanism or "Agents" in simple terms. ( I am quite sure that contagion mechanism is actually not a technical term it's actually called agents)

Even as we speak the virus is all around us. It just depends upon the immunity of a person's body to deal with the virus.

Also the virus should reach the Target Site to initiate it's incubation. In influenza virus it's the simple squamous epithelial cells lining the Trachea so you see how easy it is for the virus to cause a common cold !!!

Size of a virus is in the order of microns. So once they start incubating in your cells they form in zillilons and even more !! The cold is actually not gone "entirely" as you are thinking. Your body is still fighting the remaining ones( And thus the symptoms) but the quantity of viruses has come down to such a no that your spit, sneeze is no longer that vulnerable to others . But again if you bring a baby who has just birth and doesn't have a profound innate immunity like the elders he might even catch that virus from you and lead to a catastrophic disease were the influenza might turn up into something more deadly !!

As long as there is a symptom there is a disease It's a very basic thing that is taught to us while studying medicine