Infinity is a concept that's been used in different ways over different areas. For example you can use infinity just a placeholder for the behavior of a function that keeps increasing as you increase the independent variable(s).

But the interesting way to see infinity is found on set theory. First, you have ordinals. To get ordinals, you start from nothing, literally. You start with a set, call the empty set. Think of it as an empty bag. It's represented as {}. Now, you can get this "bag" inside another bag, let's call it {{}}. This set is now different from the first one, cause this set has something on it, the original set. You can then get these two sets into another set, to get {{},{{}}}. Again, this one is different. If you keep going, you will soon find yourself with a bunch of sets. Now, you can also order this set, by saying that a set is smaller than another if the first one is inside the other one. Because of the way we have constructed our sets, one set will either be smaller, bigger or equal to another. That's why they are call ordinals, cause they have been ordered completely.

All these sets we have made have a finite number of things inside. But what if you take all these sets and put them in another set? You will get a new set call ω. That set is basically our first infinite. But then, you could get all the initial sets, and ω, inside another set, we'll call it ω+1. You now have another infinite set!. You can keep going like this forever, and you will get infinites even larger than you could possible imagine.