This is going to be an auspicious first post, I think.
Viruses are generally pretty host-specific -- they can infect and live in (or take over) only very specific types of cells, which means that they usually can't cross species. (You don't hear about people contracting tobacco mosaic virus, for example). Some viruses, though, are less "choosy" about their host cells. West Nile Virus is one of those. WNV is carried mainly in wild birds (the birds are actually infected with WNV and will eventually die from it). Some (but not all) species of mosquitos are also susceptible to WNV and contract the disease when they bite the wild birds. Unfortunately, humans, horses, poultry, alligators and some other species are also susceptible, so when an infected mosquito bites one of these hosts, the disease may be transmitted. Not all species of mosquitos are susceptible to WNV, but enough are that it's a public health concern.
AIDS, heptatitis C and most other viruses cannot infect mosquito cells, so the virus cannot be transmitted through a mosquito bite. As common as AIDS and hep C are (and as common as mosquitos are), it's a good thing because if they could be transmitted by mosquitos, it wouldn't take long for virtually everyone in temperate or tropical climates to become infected!
Mosquitos are a lot more likely to spread bacterial infections than they are to spread viruses.