In mathematics, the remainder is the amount "left over" after division, and technically subtraction also. For example, the remainder of 10 / 4 is 2, because 4 goes into ten twice, with two left over. This is often how children are taught to handle division results that would otherwise be non-integral. You'll see this written as 10 / 4 = 2 r 2, the 'r' standing for remainder.
Of course, once we get into the realm of rational numbers, we don't have any need for remainders... do we? Well, not arithmetically speaking, no. However, they do have some algebraic use - they're the key part of Euclid's proof that there is no largest prime number, for one!
Remainder can also apply not just to numbers but to functions. That is to say, we can divide one function by another and examine the remainder. This also has many uses in function theory.
We can also find the remainder by working modulo a certain number ie, 13 mod 6 is 1, and therefore 13 / 6 has a remainder of 1. In C and related computer languages, mod is represented by the % symbol; we can find the remainder without explicity finding the quotient - that is, the real result of the division.
Remainders are generally thought of as a childish concept from arithmetic but are actually remarkably useful in proofs. I hope this was illuminating.