The amount of time you rest between sets will affect several factors that are important in the adaptations brought on by your training. The length of the rest period...
If your main goal is strength, the length of the rest intervals should be long enough to allow the nervous system to recover almost completely, but not so long that you lose what's called the post-tetanic potentiation (PTP) effect. This refers to the phenomenon by which your contraction strength potential will be increased for up to five minutes after a heavy set because of a greater neural activation.
The peak effect (greater potentiation) occurs around two to three minutes after a near-maximal contraction. Then it gradually loses its effect so that it's gone by around the fifth minute. So when training for strength, you should rest around three minutes between sets of the same exercise.
You'll still have the full potentiation effect with less rest, but you'll also have some neural and/or muscular fatigue which will counter the PTP effect. When you're doing a proper strength session, you should actually become stronger with every set of an exercise (until cumulative fatigue sets in after four or five sets).
Note that I mentioned three minutes between sets of the same exercise. If you alternate two exercises for opposing muscle groups, you can have less time between sets, provided that you still have the three minutes between sets of the same movement. For example, if you alternate the bench press and weighted pull-ups, you might do as follow:
Which would look like this:
So while the rest between sets is actually 90 seconds, you have around three to four minutes of rest before hitting the same muscles again.
By the way, the above alternating of two opposing muscle groups or movements is one of the best ways to train for strength:
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