Firstly, if your pc is under a valid warraty, you should consider taking it back for the store techs to look at. Thats what they are there for !
PSU is your power supply unit. The cord you have from the mains electricity plugs into it
Unfortunately, the best way to test a PSU is to have a spare, working PSU around, and seeing if you can reproduce the problem.
system temps are the temperatures of your different components. If parts are getting too hot, this can manifest in the way you describe.
an easy to use program that will summarise this for you is pc wizard
, just downloand, install and run, click the little picture of the hot/cold gauge for the temps