Apply Newton's second law to connected objects and systems, including strings over pulleys and elevators, to solve for acceleration and tension.
Real systems often involve more than one object linked together -- two blocks connected by a string, or a person standing on a scale inside an accelerating elevator. The strategy stays the same as always (find the net force, apply F=ma), but now it helps to first treat the WHOLE system together to find a shared acceleration, then zoom in on just one piece to find an internal force like tension. An ideal string (massless, unstretchable) pulls equally hard on whatever is at each of its two ends, which is what lets tension connect the pieces of the system together.
A classic case is two masses hanging on either end of a string over a pulley (an Atwood machine): the heavier side falls, the lighter side rises, both accelerating at the same rate, found by treating gravity on the heavier side minus gravity on the lighter side as the one net force moving the ENTIRE system's combined mass. Another classic case is an elevator: standing still, a scale reads your true weight, but while accelerating upward the scale must supply extra force to also accelerate you upward, so it reads MORE than your weight (and less while accelerating downward) -- your "apparent weight" changes even though your actual mass and true weight never do.
Mastery looks like: They find the shared acceleration of a connected system and then solve for an internal force (like tension) by applying F=ma to one piece of that system.
Common stumbles: Solving for tension by only looking at one object's equation without first finding the system's acceleration; forgetting that apparent weight (the scale reading) is a force, not the same thing as true weight.