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There's a period of time after a person is infected during which they won't test positive. This is called the “hiv window period.”
The window period can be from 9 days to 3-6 months, depending on the person's body and on the HIV-test that's used. During that time, you can test HIV negative even though you're HIV infected. You can still catch HIV from someone who is in the window period. In fact, there is evidence that a person in the window period is more likely to pass the virus on.
Antibody tests ("Rapid" tests) — give a positive result based on antibodies to HIV, not the virus itself.
Antigen tests (RNA tests) — show a positive result based on the presence of the virus. These tests are more expensive than anitbody tests, so are not offered in as many places.
Home testing kits — As of Fall 2012, there are two "home tests" which have been approved by the FDA for use in the U.S.:
PCR tests (Polymerase chain reaction tests) — also test for the actual virus. This type of test is often used for testing the viral load of HIV-positive people, as well as testing babies born to HIV-positive mothers. You can read more about PCR tests on the AIDS.gov website.