the purlple and purple/black are rear right speaker and green and green/black are the rear left speaker. that is a kenwood stereo the blue/white with ant. cont on is for an electric areil and the brown is for a phone kit this is the wire that will mute your stereo when your phones rings if you have a handsfree kit fitted.
1) wire a speaker direct to either the purples and the greens at the back of the head unit to make sure it works. if the speaker emits sound then you will no the stereo is ok.
I would then check the wiring in door looms of the car to make sure that the wires have not broken if they have then run new speaker wires from the purples and greens into the relevant doors.

2) if you have a set of multimeters set them to ohms. un plug the stereo get the plug from the car side and put the red lead on the purple and the black on the purple/black you should get a reading of about 4ohms if all is ok if you don't then you may have a wiring issue of the voice coil of the speaker may be shagged. Do the same again but with the greens.
hope this helps
if you need any more tips just ask.

P.S The red and white plugs are RCA outputs you'll have no need for these unless your using an amplifier.