HIV infected people usually do not have any symptom at first. Couple month later they may suffer a flu-like illness. There symptoms are not something extraordinary: fever, headache, fatigue, swollen lymph nodes especially in the neck and groin. Everything looks like any other flu.
Then symptoms go away. However, at this stage the affected people are very infectious. They make a lot of HIV that is easily transmitted by genital contacts. That is the danger for other people. This period of "asymptomatic" infection depends on the person. Really severe symptoms sometime occur only a decade later. Children born from HIV mothers may show symptoms earlier. Some people develop the symptoms within a few months.
During this asymptomatic period the HIV virus is actively spreading it he body. The virus kills immune cells. It also hides within infected macrophages and lymphocytes, and stays dormant. The clear sigh of HIV infection is the drop in CD4 positive T-lymphocytes counts. The virus destroys these important cells of immune system.
The loss and deactivation of the immune system eventually leads to complications. The first signs of infection are enlarged lymph nodes (lay people name them glands). This is sign of majority of infections and other inflammatory conditions. Other symptoms are also not specific to HIV only. Low energy, weight loss, fevers and night sweats are also common for tuberculosis, rheumatic diseases and many others. Later the loss of immunity leads to yeast infections (inside mouth or vagina), skin rashes, inflammatory conditions, which do not respond to regular treatment.
Eventually different weird bugs that are easily destroyed by a normal immune system, take control over in the HIV infected persons. They cause unusual infections and tumors. Herpes infections of mouth and genitalia become frequent guests. Shingles happen too often.