The max you lift in any given exercise is the process of demonstrating strength. It is just one way to measure strength, but it is one that is universally accepted. However, there are several factors that go into that one number:
- Work capacity
- Neuromuscular adaptation (i.e. the ability to recruit more muscle fibers)
- Weight x Acceleration (i.e. force)
- Kinetic vs. potential energy
I left out muscular density because at the end of the day, you can either lift something or you can't. The size of the muscle that produced the result is irrelevant to absolute strength, and only comes to play when you are talking about relative strength.
I listed skill/technique first because the ability to apply your strength into moving a heavy object requires correct technique. The heavier you can lift, the better your technique has to be to do it. This can mean the difference between first and second place if you are comparing otherwise equal people. It can also explain why someone who is technically stronger can be beaten in a fight or arm wrestling match.
Work capacity is the ability to do more work. It's hard to measure, but is probably a better representation of overall strength than an all out one rep max. It's one thing to be able to squat 315lbs for a set of 5. It's quite another to do that 10 times. When you look at how elite strength athletes train, they stay away from their maxes and focus on increasing their work capacity.
Neuromuscular adaptation is the ability to call more of your muscle into doing work at once. When you are untrained, your body only uses a small portion of your muscles to do work. This limits the all out max you can do, but saves your muscle for endurance. As you progressively get stronger, you can make increases in strength even when you are not increasing muscle size or hypertrophy simply by getting more of your muscle to work together. It's a natural byproduct of increasing weight over time.
Force is the purely physics related portion and is simply weight x acceleration. In short, the faster you move the bar the more force you produce. Similarly, the more weight on the bar the more force you have to produce to move it.
Kinetic vs. Potential energy describes whether the weight being moved is in motion or is stationary. A 315lb barbell sitting on the ground has more potential energy than a 225lb barbell. It requires more force to turn that potential energy into kinetic energy. A weight already in motion is easier to move than one that is stationary.
Fatigue vs. Adrenaline describes the effect of your endocrine system on your ability to lift on any given day. Heavy lifting and high volume lifting increase fatigue. Fatigue is like running with an open parachute into high winds. It severely limits your ability to demonstrate strength, but it is a normal and necessary part of getting stronger. In contrast, with a proper peaking cycle, you can ensure fatigue is at a minimum when you go to test your numbers in contest. Combine that with the adrenaline rush of performing that lift on stage, and you can perform well above your normal every day level of strength. That's a big reason why someone with a 1000lb deadlift can do it on competition day, but a couple days later wouldn't be able to come close.
The best measure of strength is your every day max. This will be far below a test day or competition max. It's what you can do even when you are very fatigued, and your technique is less than stellar. Increasing that every day max will have a direct result on increasing your competition max.
If you can curl 40lbs today, and someone you are challenging can only do 35lbs it might be just because they did 50lbs the day before. Or it might be because of life's stresses getting to them. When you get into a competition like arm wrestling, they may be able to best you because they can call on more of their reserves or adrenaline has a bigger effect on them. Or they simply know how to bring an arm down better than you.
There's a lot that goes on with getting stronger. Demonstrating strength is its own discipline. Olympic weightlifting, strong man events, and power lifting all have different requirements. You might be really good at one, and really bad at another. But that is likely due to technique more than just being weak.