The commonest problem where the light is on but there is no cooling is a broken or sticking thermostat.
Locate the knob inside the fridge that controls the temperature (the thermostat is connected to the knob) and rotate it fully in both directions a few times to see if that will "unstick" it. A few gentle taps my free it too.
On my last go I got a dodgy sounding steady power hum (louder than the normal fridge noises) so I left it like that overnight. This morning it was still making the same sound but no cooling still, so I have turned it off as the sound concerns me.
OK, so it sounds like the start relay is broken/stuck or the compressor is knackered. If you know what I am talking about - you could check the relay and reseat it (give it a shake too - if it rattles it's knackered).
Replacement relays are cheap, compressors are not and usually it makes more sense to replace the fridge.
Don't want to really to rub it in But...I bought the recommended best buy by Which with a tax rebate in 1980/81, Electrolux Fridge Freezer, double compressor, can not even hear it running even now, in fact one of my brothers who specialised in ice machines at that time did not even think it was switched on...never even had to change a fuse and both lights have never even needed new bulbs!
reading that back it implies smugness! just meant to re-inforce how well made things used to be! :-)