Most often, a technician or doctor will draw blood from your vein and check it to see if there are antibodies for HIV. You can also test other body fluids -- oral fluid (not saliva, collected from your mouth using a special device) or urine -- but these aren't as sensitive or accurate as traditional blood tests. Some rapid screening tests can give results in 20-60 minutes.
Current blood tests can find both antibodies and a part of the virus itself. These could give a positive result as soon as 3 weeks after HIV exposure.
Home testing kits found at drugstores are really home collection kits. You prick your finger with a special device, place drops of blood on a specially treated card, and then mail the card in for testing at a licensed lab.
If any of these screening tests say you're positive, follow up with a doctor and more testing to confirm it.