Calculus ("Infinitesimal Calculus" or "Differential and Integral Calculus") is about calculating local differences and global sums (of the values of functions) using the concept of infinity.

The best way to intuitively and comprehensively grasp this is to learn Calculus in this order:

Algebra -> Functions -> (Calculus starts here) Finite differences -> Finite sums -> Infinity/Infinitesimals -> Epsilon-Delta

Unfortunately, it would be an understatement to say that pretty much no calculus learners learn it that way. No wonder why calculus stays so foreign to people. They are mostly trained to get A's by mechanically calculating values based on memorized tables, all while completely missing the bigger picture.