1. Most likely is discharged or improperly inserted batteries. Make sure to use freshly charged AA batteries that are inserted with the correct polarity. The Inner 2 have the minus side up and the outer 2 batteries have the + side facing up.
2. Close the battery door fully because the battery cover pushes on a small sensor that has to be closed before the camera will turn on. The battery cover slides and clicks into place. If the battery cover sensor is bent or damaged then the camera would not turn on. The battery cover sensor is tiny, and out of the way and probably not obvious that is has one.
3. Insert the proper memory card and close the memory card cover securely. If it's left open or not seated fully the camera may not turn on. I
4. Also remove the memory card and re-seat it, then close the cover again in case it was not fully closed.
5. Remove the batteries and check that the battery terminals are clean and they appear to be intact. For example, corrosion from leaking batteries could cause a poor connection.
6. Was the camera dropped? If it was dropped that can bend the metal battery contacts, then when you put the batteries in it doesn't really contact the battery properly.
7. If it still won't turn on have an electronics guy check it over, or at the local college electronics teacher. I've fixed these before and the main DC circuits are not too complicated. These camera's can even be powered up using a regulated 6V DC power supply connected properly to the battery terminals, (just for testing) and in that case you don't need batteries to tell that the camera works or not. These draw about 300mA of current when ran from 5V to 6V DC.
I attached a JPG photo showing a similar Canon A60 running via a DC power supply and no batteries. This one was apart at the time to fix a stuck zoom lens, and when it's apart there is no where to put the batteries so I have to power it this way. You do not have to take it apart
or use this method. I just mentioned this so it's clear that all it needs to run is 5 -6V DC from properly charged and inserted batteries (or an external source). Don't try connecting anything if you are not familiar with electronics because connecting it up wrong can damage the camera. Try the suggestions mentioned early and it will probably work since these are pretty reliable cameras.