Viruses such as HIV infect cells by binding with the DNA within them. Biologists tend to describe cells as being like tiny factories, which is probably a good starting point - the double-helix of DNA splits into two singles, each being essentially able to recreate the double on account of the fact that each of the four basic parts (referred to by the letters A, C, G & T) will only combine with the correct partner. In this way, the nucleus of a cell (which is where the DNA is found) produces 'copies' of the DNA in order to produce more cells of the same kind.
The virus particle enters the cell and is attracted to the nucleus (this does not happen in red blood cells as they have no nuclei). By releasing its choroid coating (or 'shell' if you prefer), the DNA of the virus attacks and replaces that of the host cell and thus the cell is set to replicate this DNA instead of its own.
Extending the 'factory' analogy, the cell is now set up to create products on behalf of the invader rather than on behalf of the original 'customer'. The nature of the 'factory' has been changed.
Virus DNA is generally more rapidly reproductive than that of the invaded host cells, so a great number of virus particles are produced before the invaded cell dies - and the number of virus particles within an organism grows very quickly.
Since the alterations to the DNA structure of the cells of the infected organism would mean the function of the cells is invariably affected, this effect is what we interpret as a disease. Sometimes the affected organism (be it a man, a dog, a worm or whatever) is capable of detecting the change and eliminating the 'rogue' cells. This is the function of the immune system. If the organism cannot do this, the virus has won the battle - and the organism dies.
I hope this helps.