Infinity is the word we use to represent something that has no limit, edge, boundary or final value. It is something which doesn't finish or lasts forever. The best example is numbers. The biggest number you can think of can still be multiplied by 2, or have another number added to it. The scale of numbers doesn't come to an end so we just keep counting and counting and counting forever. Infinity isn't really a number because you can't do simple maths with it. For instance, if you take an infinite set of things and add one, you have gone from an unmeasurable number to another unmeasurable number i.e. infinity + 1 = infinity again. This makes it more of a concept than an actual number.
Where it gets really interesting is when we consider different types of infinity. Take the set of whole numbers (natural numbers) 1, 2, 3, 4... we end up with a set that goes to infinity. But now imagine we did it with only the even natural numbers 2, 4, 6, 8... we would still get to infinity. Thing is, the normal numbers are (even numbers + odd numbers). There are half as many even numbers as normal numbers. So the infinity we count to using the even numbers is half the size of the other infinity.
Or, simpler version, imagine we started at 2 and then "counted to infinity". 2, 3, 4, 5 etc. etc. This will still get to infinity, but we haven't included the number 1, so this infinity should be smaller than the other infinity by a whole number. Or imagine we counted all the decimal points. 1, 1.1, 1.2, 1.3 and so on...this infinity would be bigger than the normal infinity.
A man called George Cantor set out some rules for dealing with infinities in the late 19th Century. He referred to the full set of natural numbers as "aleph null" and then other sets of infinities as "aleph one", "aleph two" and so forth. There's a whole branch of mathematics dedicating to dealing with infinities called "transfinite mathematics" which deals with these alephs.
Interestingly, some mathematicians don't believe in infinity. Doron Zeilberger, for instance, is convinced that infinity is a mistake and that if you count up the numbers and keep going you just loop back around to a smaller number eventually...there really is a biggest number out there somewhere!
From a Scientist's perspective infinities are very useful because there isn't a way we can see for them to exist. Infinity contains all sorts of logical paradoxes and impossibilities which can be written on paper but translate to impossibilities in reality. So we use infinity as a kind of test: if a hypothesis contains an infinity somewhere, it's incomplete or we've made a mistake. Only once we've removed the infinities, can we be confident we've got a theory that works.