The optimal amount of time you should rest between sets of weight exercises is very much dependent on the type of exercise being undertaken as well as the effort being expended.
I have found that for the determination of rest periods, the three categories of training can be listed as Strength / Endurance & Size / Fat Burn.
If you are actively looking to build up your strength, then it is likely that you will be concentrating on shifting heavy weights with low reps, so for example, on the Bench Press, you would be looking at a rep range of 6-8 with weights that your muscles are failing at beyond this range. With the greater stress being placed on the muscles, and with the requirement for those muscles to be called on again for the second and third sets, the key is recovery. Giving your muscles a 2-3 minute rest (even as high as 4 mins for heavy compound lifts such as the Dead lift, Squat and Bench Press) is an optimal amount of time for the muscle to be ready to lift again. Any more than this and you run the risk of the muscle getting cold and unable to fire at the effort required.
If your aim is to build size or increase endurance, then generally you will be looking at shifting medium weights for higher rep ranges, so around the 10-12 range. Whilst still putting large stress on the muscles, the reduced weight means that the muscles can recover more quickly than at the lower rep/higher weight range. Aim for around 1 minute rest here.
Finally, for fat burning, the routine is likely to be lower weights for high reps 15-20, even up to 50 reps for pyramid sets. You may also be using super sets to increase metabolic rates. This crosses over with the principals of HIIT where the key is to keep the heart rate high for short periods spaced with short periods of rest. Aim for around 30-45 seconds here.
Follow these guidelines and see how you get on. If you are struggling on the proceeding set, add an extra 10 seconds to the rest period and see how you go. These guidelines are just that, guidelines, and the optimal rest period for you may be slightly different.
Good Luck – and remember Health and Fitness is Doable.