The folk understanding of strength is that we lift heavy things, get bigger muscles, and are stronger. If we were just kids running around and playing and lifting things for fun, that understanding would be plenty, but if we are going to seriously train, we need to know more than that.
When you first begin training, both strength and hypertrophy come fairly quickly, but it is wrong to assume that all the strength comes from hypertrophy. Just like a complex skill can be neurologically grooved, so can the simple skill of how to lift something heavy over your head. So much of the initial gains in strength are actually neurologically based.
We sometimes talk about "nerve force"; this is basically a way of describing neurological connectivity. As we do an activity more and more, the neural pathways that participate in the signalling for that activity get mylenated. Myelin acts as an insulator (literally) on nerves cells, so that nerve pathways that are more heavily myelinated carry signals faster and "harder" than other nerves. This is so called muscle memory.
Just as effective stretching is convincing the muscles to relax more, effective strength work is largely convincing the muscles to contract harder. More muscle fibers will generally mean a stronger muscle, but this is why a little guy who is well trained can out lift a big guy who isn't (watch the Olympics and see if you think you can squat as much as a 135 pound lifter).
We all basically know that bigger muscles make for more strength. But its not quite that simple.
There are two kinds of hypertrophy (muscle growth): myofibrilar and sarcoplasmic. When you do low rep, high intensity training, you are encouraging myofibrilar (muscle fiber) growth, and there will be commiserate strength gains. Sarcoplasmic hypertrophy is encouraged by the kind of training where you "feel the burn", high rep, low weight, low rest; the sarcoplasm is a kind of goo that houses fuel for your muscles. Sarcoplasmic hypertrophy is important for muscle stamina, but it will do nothing for strength. If you see somebody with huge, bulging muscles that aren't that well defined, or notice that the person looks stronger than they really are, then they've spent too much time on high rep, low weight exercises.
I just like the word sinew, but what I mean here are your soft tissues, including ligaments and tendons, but also the actual structure of the muscle and tissues surrounding the muscle. Muscle is extremely vascular (lots of blood goes to it) and so responds relatively swiftly to training. However tendons and ligaments are not nearly so vascular and receive much less blood supply, which slows their adaptation. You should expect adaptation of tendonous tissue to take six times longer than muscle. This is why it is a mistake to add more and more and more weight every week, which is a huge temptation for new trainees who see such amazing gains in the first several months. Time must be spent to stabilize the strength gains, allowing the soft tissues to strengthen enough to keep up with the increases in weight.
Rushing an increase in weight is asking for a soft tissue injury, but it is also counterproductive. A great deal of our strength comes from the soft tissues, both in their elasticity and in allowing them to get strong enough so that the body can "let off the break" and allow for greater exhibition of strength without fear of injury. By elasticity, I mean literally the elasticity of the tissues, their ability to stretch and contraction. Imagine a runner bounding across a field; the speed doesn't come from huge amounts of flexing the muscles and other soft tissues, but rather the springiness of the muscle lengthening and shortening in an extended position. Regarding "letting off the break", the body has mechanisms besides pain for understanding injury or danger of injury. Many people experience a sudden feeling of weakness preceding a tendon injury after they push through it; the perceived weakness was actually the body's way of warning the brain of damage to the tendon. By slowly strengthening the soft tissues and showing the body that it can exhibit strength without injury, the body "takes off the breaks" and allows for further strength development.
The soft tissues need both heavy loads but also constant movement to develop well.
Bone is the frame upon which all our muscles rest, and if muscles grew incredibly strong without the tendons and bones being given time to also grow strong, then the muscle could not only rip itself from the bone, but could snap a bone in half. In fact, when electrocuted, the muscles all contract as hard as possible, and injuries do occur where the muscle rips tendon and bone apart.
Bone takes even longer to train than the soft tissues; one should estimate twelve times longer than muscle. Like soft tissue, it needs heavy, frequent stress for adaptation, but it also recovers much more slowly. The truth is that usually people in training overload the soft tissues long before they overload the bone, but as people age keeping up bone density becomes increasingly important. Bones that are made to support weight have more collagen, which makes them not only stronger but also springier and less brittle.
This should give you a basic understanding of some of the different adaptions the body must undergo to gain strength. The takeaway here is that when you are training for strength, you are not just trying to make your muscles bigger. The neural adaptations are very important and the closest to limitless in their adaptive capacity; if we had to rely only on hypertrophy, you wouldn't have 132 pound guys squatting 600 pounds. We also have to be aware of how our training is signaling our muscles to adapt; some sarcoplasmic hypertrophy is good (and unavoidable anyway), but if strength is our goal we shouldn't be chasing high reps at low or moderate weight. And finally, we can't get so far ahead of ourselves in training that we do not allow time and rest for the soft tissues to catch up with our muscles; doing so makes us more resilient to energy and also stronger.