An infinite set is exactly what its name suggests: a set with an infinite number of elements.
One example that you'll no doubt be familiar with is the set of natural numbers, N. Also called the "counting numbers," the natural numbers are the positive whole numbers 1, 2, 3, etc. Since there is no "last" natural number, N is considered an infinite set. More specifically, it's what's called countably infinite. More on that in a bit.
Another example of an infinite set is the set of real numbers, R, which includes all of the naturals, integers, rationals, and irrationals. R is also an infinite set, but it turns out, shockingly, that it's a bigger kind of infinity; uncountably infinite, to be precise. We say that a set is countably infinite if every element can be matched up with a natural number - in other words, if there's a way the elements can be lined up and counted one after another in some way. This is true of the integers and rational numbers (and I've attached one way you can "count" the rational numbers in order), but not the irrationals.
