Highly likely that a rail has blown in the power supply unit, and now it cannot provide enough watts / amps to start then run the computer. unfortunately PSU's often take other components with them when they go, but check out if you can get a replcement, actually measure the PSU box dimensions, then look at the information stuck on it, and then try for a replacement do not jump and get one as cheap as possible or you will just get the same problem if not worse again very soon.
named brands Zalman, Tagan, Hiper, Enermax, Cooler master, Seasonic, Akasa, Blue Storm, Jeantech