Strength can be simply defined as ones ability to produce force. If you still doubt that being stronger makes you a better athlete, read this, or this, or this... Have I made my point? Good.
So what does 'ability to produce force' actually mean in practical settings and how can you train to be better at it, producing more force and doing so more quickly.
Using the force
There is a relationship between muscle size and the amount of force you can produce. A bigger muscle has greater potential to provide a bigger output of movement. But if size were the only factor, then top level body builders would be winning powerlifting, strongman and weightlifting competitions. And they'd do so in a thong, because that's how they roll.
The other, and arguably more important factor for us as athletes, is to use the muscle we already have to its fullest capacity. We can enhance our central nervous systems ability to recruit muscles and thus the force we produce is increased.
Why don't we just get bigger?
Carrying more muscle mass comes with a metabolic cost. Muscle uses energy and therefore both oxygen and glycogen. To fuel more muscle mass we need to carry more glycogen and/or take on more oxygen more quickly. This affects our cardiovascular fitness.
Another reason is that muscle mass (or weight in general) increases our ground impact forces during running, changing direction and so on. If we carry more muscle than we really need to be effective then we are potentially slower moving.
Not to mention that many of us compete in strict weight categories so want to be as strong as possible within a certain weight class. If we are lean (i.e. not carrying lots of fat mass that doesn't help us compete) then there is a limit on how much muscle we can gain without moving up into the next weight group where we may be less well equipped.
The key point here is: we want only as much muscle as we can be EFFECTIVE with. If we add muscle, it has to bring us benefit that we can't get from our central nervous system. That could be being heavy enough to wrestle other heavy guys, or to serve as protection to joints and balance the body's strength ratios. Otherwise you'd be better off recruiting your existing muscle not building more.
How do you recruit more muscle?
As stated above, the thing that determines how much of your muscle that you can 'access' is the control system - the Central Nervous System (CNS). Made up of your brain and spinal chord, then the peripheral nervous system - made up of the other nerves stemming off the spinal chord delivers the signals from your brain to the muscles.
Making your brain send bigger, more frequent signals is essentially what we are after. We do that by lifting heavy things that challenge our CNS to provide enough of a signal quick enough and we repeat those efforts to stress the system. Only when we've stressed the system will we get a change in our ability. No stress means no adaptation.
What is an appropriate stress to get stronger?
For the tension to be adequate to stress the CNS, you need to use loads that only allow a few repetitions of a movement. If you can bang out 20 reps then the load is simply too light to place enough stress on the CNS. It's not useless - it trains and improves endurance - but IT ISN'T STRENGTH.
Strength requires loads that allow only 1-7 (ish*) reps. It means that you literally cannot do another rep, then when you've had 2-3mins rest you do another 1-7 reps. How experienced you are and how strong you are, where you are in relation to competition and your season will all influence whether you did 1-3 sets of 1-3 reps, or 3-5 sets of 3-7 reps.
This kind of training is heavy, and because it's heavy it is usually slow. Rest periods are long because your CNS takes a while to recover enough to repeat the same maximal efforts. A session could last 30-120minutes depending on your level. A newbie could get stronger doing 30mins of strength training - only completing 3 sets of 5 reps on 2/3 exercises. An elite weightlifting may require more like 6-10 sets of 1-5 reps on 4/6 exercises.
How often can you train for strength?
Again, depending on how advanced you are, you could train as often as twice daily or as little as once every 5 days. Only a real novice can get away with once a week strength training and you'll quickly need to move to once every 5 days as a bare minimum. The structure of your training can influence this too. If you were to squat heavy in one session but deadlift heavy in the other, you could fit 2 strength sessions into a 7 day period, or even a 5 day cycle. The key thing being that strength is specific. You only improve what you train (to a certain extent) so if you're smart with exercise selection you can plan a training week with lots more volume than if you were to try to do the exact same thing each session. (ref)
Exercises use a motor pattern - a way of moving in coordinating sequences - which is temporarily harmed from training stress. Through adapting to the stimulus (your strength session) you improve the synchronicity of inter and intra-muscular coordination (ref). If you train a different movement pattern therefore, you can stress something different and not have too much detrimental effect on each other. This is one way some people can fit in 4-5 strength training sessions into a week without decreasing in strength.
What about people who train twice a day and often do the same things each time they train?
There are a few exceptions to the basic rules. Elite athletes have built up their training tolerance over time so that rather than one single session being enough of a stimulus for change, it takes a concentrated block of training to make them stronger. Examples of this can be seen in weightlifting and strongman sports where lifters may train the same (or very similar) movements several times a week. Here though, the planning and periodising of training volumes must be carefully planned and recovery strategically placed.
Chances are you are not an elite level lifter. Yeah, me neither. You're programme should be applicable to your situation. A little trial and error will soon show you what you respond best to. In most cases, research shows 2 sessions per week on top of sports training provides enough of a stimulus to enhance strength and ultimately, performance (Rhea et al., 2003). If you're getting stronger and performing well with 2 sessions per week, I wouldn't add 3 sessions just for the sake of it. If improvements slowed down or stagnated, adding a 3rd session would be more justified.
To see what a strength session looks like check out this example programme. For to understand how much of it to do each time, read this blog post. For more on how to structure your training week read this blog post.
To help you figure out where you are in the strength training continuum, and to help you plan sensible progressive programmes, I wrote Starting out - Beginners guide to S&C. Don't waste time on loads and rep ranges that don't get you where you want to be.
Rhea, M. R., Alvar, B. A., Burkett, L. N., & Ball, S. D. (2003). A meta-analysis to determine the dose response for strength development. Medicine and science in sports and exercise, 35(3), 456-464.