What does closure in algebra mean?

When talking about closure, you have to be careful to specify two things: the set, and the operations. The set is very important, because there are operations for which some sets are closed and others are not.

The set of integers are closed under addition, multiplication and subtraction. What this means is that if x and y are integers, then x+y, x×y and x-y are also integers. The integers are not, however, closed under division, since x/y isn't necessarily an integer. For example, 2/3 and 5/0 are not integers.

The natural numbers are also closed under addition and multiplication, but are not closed under subtraction. For example, 2 and 3 are natural numbers, but 2-3 is not.

The "closure" of a set S under some operation is the smallest set which contains S as a subset and is closed under that operation. So, for example, if your set is the natural numbers, and your operation is subtraction, then the integers are the smallest set which a) contains the natural numbers, and b) is closed under subtraction.