The best and most effective changes are repairs: a crack or tear can be made to vanish, in many cases a missing bit (providing it isn't too large!) can be reinstated.

The overall quality of the image can often be improved EG the focus crispened up, the contrast range increased. But, overdoing these sort of changes can spoil the photograph, so I'd recommend caution. Just because it can be done doesn't mean it should be.

In most cases annoying blemishes, be it they physical ones on the photograph, or something in the original composition - EG the classic 'tree growing out of head' - can be removed. Distracting backgrounds can be altered or defocussed. Detail lost in darkness can be enhanced.

In some cases it may be possible to change the composition significantly. The classic example for this is to 'air-brush' the ex-boy/girlfriend out of the holiday snap!

Note: currently, the biggest photograph I can work on is A4 size (8¼ x 11½ inches).