When a person talks about strength, one usually thinks about bodybuilders or power lifters (they aren't the same). Or at least I do. Strength is a measure of how much force a muscle, or group of muscles can deliver in some specified way. It takes strength to stand up, to walk, to carry groceries, to dial the phone.
This page will talk a bit about what happens to the muscle on a cellular basis, from there we will move to some training regimens. Nothing here is absolutely known by myself. Most of this is my opinion of things that I have read or experienced.
A muscle is made up of many cells, some of which are motor cells. These are cells that are capable of decreasing their length when stimulated by a nerve and in the presence of sufficient ATP (adenosine tri phosphate). Practically speaking, each motor cell has only 2 lengths, a relaxed length and a contracted length. Intermediate lengths are not allowed.
So for a person to lift a weight (perhaps a shopping bag), the overall muscle has to shrink (and also get thicker). It does this in a largely cooperative manner: first one motor cell contracts, then another, then another; until enough motor cells have shrunk to decrease the muscle length the required amount. The time periods between one cell (or perhaps a small number of cells) contracting and the next cell or set of cells contracting can be quite short.
If the muscle doesn't have to generate much force, then at any given time not many motor cells are trying to contract. But if a lot of force is required, many motor cells (maybe even all the ones that haven't yet contracted) may be attempting to contract. This fraction of cells trying to contract is something quite manipulatable by training.
So there is some maximum amount of force which a muscle can develop. This maximum is affected by diet (we need ATP, among other things), training (the nerves have to fire) and geometry (how long bones are, where muscle attach, what we are trying to lift, the position of our body). Our bodies are made to adapt to chronic demands for strength, by building up the muscle. Increasing how much force each motor cell can exert in trying to reach its contracted position.
The construction of our muscles also allows for an elastic component. Our muscles are capable of acting somewhat like springs. If we "bounce" weights, especially at high speeds, the muscle is almost entirely acting like a spring. We may damage cells by overloading the elastic components, or we may damage the places where the muscles attach to bone. In any event, the type of stresses which come from bouncing weights is seldom what is required to build strength.
Our muscles suffer from a phenomenon called internal friction. Because of this internal friction, the maximum force a muscle can exert depends on whether it is getting shorter (concentric contraction of the muscle), staying the same length (isometric contraction) or getting longer (eccentric contraction). The maximum force which can be developed for a concentric contraction is less than for an isometric contraction, which is less than that for an eccentric contraction.
Or in a nutshell, we can lower a heavier weight than we can lift.
On the strength scale, we have 3 well known types of strength: