What's the most important component of physics that's mandatory to understand/remember?

Study and understand the physics; don't memorize a bunch of equations. Once you understand the physics; so you can explain it to your buddy or the teacher, you can always derive whatever equations you need.

We see it over and over in Answers where the Asker says, I have the equation, but have no idea how to use it. That's because the Asker has not learned the physics. It's the physics that guides us how to derive and then use the equations.

And a major part of learning the physics is in learning the definitions. For example, learn the definitions for force and energy; so that you can tell us what the difference is and how they interrelate. If you can't tell you buddy or the teacher what force and energy are in plain language, you'll have little chance in solving problems about them.

Mathematics. It's all about the mathematics. There are actually two complementary pieces to this -doing the mathematics and the ability to see the mathematics as physical process.

That might not be the answer you're expecting.

At the beginning the mathematics is simple - arithmetic, basic algebra, a bit of trig. It does get more complex along the way - calculus, differential equations, and other stuff.

But you also have to grasp what the mathematics represents, otherwise you're relegated to trying to plug numbers into equations.

For example, one of the basic equations of motion is d = d(initial) + v(initial)t + 0.5at^2. But you have to understand how displacement, velocity and acceleration are related in order for the equation to be useful.

[I agree with oldprof. His answer included what I meant to say, and I think we're on the same page, but he did a better job with it.]