Viruses hijack your cells, making them make replications of the virus that invaded the cell. 

A story of a stray virus:

You were just eating in the cafeteria. It was flu season, but you didn't mind. Eating in a table full of people because your friends were there, you were exposed. Someone sneezed. You didn't see it, since the droplet was so small, but some snot dropped onto your fork. Not even taking notice of the person sneezing, you kept eating and when you were breathing, you inhaled the virus.

You didn't feel so weird at first, but everything started late at night. 

Deep in your lungs, the virus had started its job. Slipping by unnoticed by the immune system, it goes to a random cell. Sending chemical signals to the cell, it tricks the cell into letting it in. Enclosing the virus in a protective membrane, the cell pulls the virus in. The virus suddenly takes control of the cell, hijacking its code. Changing the protein structure used to make a new cell, the virus turns it into the "diagram" of a virus instead. Meanwhile, T cell "patrols" come over and check. Through the MHC l molecules, they see different proteins than what should be there. Alarmed, the T cell orders the cell to destroy itself by apoptosis, the programmed death of a cell. The infection is over.

Now, let's go another way, pretending that the virus was not seen. Then, it makes copies of itself using the cell's resources until the cell bursts, releasing thousands of viruses (They sure are tiny...) into the body, where they will either die or make more viruses. This repeats until dendritic cells, killer T cells, and natural killer cells arrive to stop the infection. 

"How do they stop the infection?", you may ask. It is simple- the dendritic cell goes to the lymph nodes, as explained in here, and finds the right T cell, then the activation process begins: some helper T cells go to the battlefield, encouraging and ordering warriors, and some go to activate killer T cells and B cells. When killer T cells get activated, they go to the main infection point and search for the antigen, and B cells and plasma cells produce antibodies.

This is why an infection might last a week; this process may last three to four days, and the ending battle may last three to four days.