Different weight levels, repetitions and sets cause your muscles to develop or "adapt" in different ways. Here's a rundown of what these are all about, we'll review the way to train for these specific goals in the article below.
Strength is the ability of your neuromuscular system - your brain, nervous system and muscles working together - to generate force. A big part of strength is neurological. (We've all seen someone who doesn't look very big but is very strong).
It is helpful to think of two kinds of strength. Maximal Strength is the type of one repetition strength that powerlifters train for. Endurance Strength is the ability to produce force over prolonged time, important for marathoners and most of the rest of us.
Hypertrophy means getting bigger muscles. Fast twitch muscles fibers typically hypertrophy more than slow twitch or endurance muscle fibers. Weight training programs for those seeking hypertrophy need to target the fast twitch fibers. There is also a neurological component, as the nervous system needs to establish a connection to the maximum number of muscle fibers in order to get them working and growing.
Power applies to generating the maximum amount of force in the shortest period of time. The brain, nervous system and muscles work together to hit a baseball out of the park. Training for power generally involves quickly moving both heavy and light loads.