How does a person get AIDS?
You read right: it can only come from people already infected through:


sexual contact
contact with an infected person's body fluids (blood transfusions), although not all fluids carry HIV (saliva, tears)
from mother to child (pregnancy, breast feeding)


Having unprotected sex with multiple partners statistically increases the risk of stumbling upon someone infected.

A person carrying the HIV virus may not show symptoms of AIDS, because the HIV infection passes through a stage of clinical and symptomatic latency (the asymptomatic HIV or chronic HIV) which can last up 20 years (average of 8 years) when only uncommon symptoms like fever and weight loss show up. 

Where it started? It is believed that HIV originated in non-human primates in West-central Africa and were transferred to humans in the early 20th century. The virus adapted and became strong, because the virus from primates is typically suppressed by the human immune system within weeks of infection.

HIV is a retrovirus that primarily infects components of the human immune system such as CD4+ T cells, macrophages and dendritic cells. It directly and indirectly destroys CD4+ T cells.

So it destroys the cells reponsible of its removal from the human body. Hard to believe that there are people whose immune can really destroy the virus. A virus lives inside the cell it parasite, so the immune cells must destroy one the other. Also the virus infects and kills them (the good ones). However, as MattDMo suggested an infected person may take antiretroviral drugs and the disease progression can be slowed down.

Source: Wikipedia - HIV, AIDS