As part of its lifecycle, HIV integrates its viral DNA into the DNA of the cell it infects. In a normal infection, the viral DNA is then processed by the infected cell's own gene expression machinery and the virus starts to replicate. However, sometimes instead of the virus being expressed and made, the DNA is "silenced" by the infected cell, meaning the viral DNA is there but not being actively expressed by the infected cell. These cells then harbor the latent virus for as long as these cells are alive, which for some memory immune cells can be for the rest of your life. This is the virus reservoir. If you take anti-retroviral therapy (ART) drugs, such as the NNRTIs mentioned before or protease or integrase inhibitors, these will inhibit active viral replication, but won't cause any harm to the reservoir viruses that are latent. Randomly* as well, these "silenced" virus DNAs in infected cells that make up the reservoir can become un-silenced, and the virus will start replicating. If you are still taking ART, then nothing happens. If, however, you stop taking the medication, these viruses that pop back up will re-start the HIV infection and within a few weeks you will be HIV+ with viral loads (amount of virus in your blood) the same as before the ART treatment was started. This is why the ART medication must be taken for the rest of the patient's life, not because big Pharma wants to make extra cash.
Interestingly, if you follow patients that have lapses in their ART treatment and sequence the viruses that repopulate the infection, they become more similar (clonal) over time, due in part to the reservoir cells! Since potentially a single virus will do the repopulating from a reservoir cell, you would expect the resulting population of viruses to be more similar to each other than in the original infection, and this is what is observed: Specific HIV integration sites are linked to clonal expansion and persistence of infected cells
* Random by measurement, not necessarily by mechanism.