Good question.
I'd say Calculus is a very useful branch of mathematics which lends itself very well to real physical problems. Common problems we can solve using calculus might be falling objects or the orbit of planets and moons, where our variables are time, velocity, distance, acceleration, etc.
I think it is important to note that Calculus is rigorous. By that I mean that we have some definitions and axioms to start with and then we can build all of calculus.
Before calculus, we could solve many if the same problems that we do today, except that today it's much more satisfying. Back in the day, the Greeks would use exhaustive methods.
Here is an example. Imagine a circle. Just for kicks we want to calculate the area of that circle with squares. We put one big square in the middle so that the corners touch the circle. Well, there is a lot of space not covered by the square. So let's use smaller squares. So then we use squares that are half the size and fill in the square. Still we haven't covered all of the circle. So we use smaller squares, and smaller and smaller...
But there is always some space left around the edge of the circle. Very frustrating!
And then when we are exhausted from calculating how many tiny squares will fit in the circle and have added up their areas, we give up. This is the exhaustive method. I jest.
But seriously, when we fell we are close enough we stop.
But "close enough" is just not acceptable to mathematicians. So Newton and Leibniz developed calculus. they developed a method that would allow us to continue to add up the tiny squares until their size becomes zero (we often say infinitely small or arbitrarily small).
The foundation of calculus is the limit. In the problem I stated above, we would look at the sequence of areas produced by the squares. We notice that these approximations get slightly bigger and bigger (closer to the actual area of the circle). And we take the "limit" of those approximations as those square become zero in size.
make sense?
