When people ask me what calculus is, my usual answer is “the mathematics of change,” studying things that change continually. Algebra is essentially static, studying things frozen in time and space. Calculus studies things that move, shapes that bend, etc. Algebra deals with things that are exact and consequently can be fragile. Calculus deals with approximation and is consequently more robust.
I’m happier with the paragraph above if you replace “calculus” with “analysis.” Analysis certainly seeks to understand and model things that change continually, but calculus per se is the mechanism of analysis.
I used to think it oddly formal for people to say “differential and integral calculus.” Is there any other kind? Well yes, yes there is, though I didn’t know that at the time. A calculus is a system of rules for computing things. Differential and integral calculus is a system of rules for calculating derivatives and integrals. Lately I’ve thought about other calculi more than differential calculus: propositional calculus, lambda calculus, calculus of inductive constructions, etc.
In my first career I taught (differential and integral) calculus and was frustrated with students who would learn how to calculate derivatives but never understood what a derivative was or what it was good for. In some sense though, they got to the core of what a calculus is. It would be better if they knew what they were calculating and how to apply it, but they still learn something valuable by knowing how to carry out the manipulations. A computer science major, for example, who gets through (differential) calculus knowing how to calculate derivatives without knowing what they are is in a good position to understand lambda calculus later.