Calculus is a branch of mathematics that allows people to more accurately calculate what is going on with functions and graphs. It is not simply derivatives and integrals, however those are big parts of it. For me it's hard to relate what it is, beccause it's like asking what algebra is, it's many things. Most people think of derivatives and integrals when thinking of calculus, and the simplest explanation for those are that derivatives allow you to get data from a function at a specific point, and integrals allow you to calculate the area under a curve, no matter what that curve looks like in many cases. There are some curves that you can't calculate an integral for but that's a long and complicated discussion. Physics uses calculus to figure out what is going on with an object/thing/particle at an exact instant, by taking the derivative of the function describing the motion. Business uses calculus to optimize costs and productions by examining graphs and finding maximum and minimum points in the graph. you can also use calculus to get the volume of objects that algebra can't deal with very well, i.e. donuts. For example to calculate the volume of a donut you would have to assume that the donut is spherical for algebra, and then stretch it out to a cylinder, what happens if the donut is not perfectly spherical though, what if there are ridges, gaps, etc. Calculus allows us to take these things into consideration with usually just one or two equations. Where algebra would require that we construct different volume equations for every irregularity in the surface, and even then wouldn't be very accurate.
This is a ver very short description of what calculus includes, kind of like saying a computer has a resistor in it. If you'd like to see an article on this I can see what i can come up with that's more informative.