∞ is a symbol that shows up all over the place, and it is called infinity. You can sometimes do operations on it, like numbers, and you can give equalities to it, and you can take limits. It's not always a number, but sometimes it acts like one.
When pressed to give a definition to ∞, I said "∞ is the symbol which, for any number system with numbers n and an ordering relation >, it is true that ∞ > n". "number system" is left ambiguous, and is very broad - it covers naturals, reals, surreals, cardinals, second cardinals, and whatever else you think of.
Now, this definition itself is somewhat strange; number system is such a big concept that it may be that you can't compare between them. Even so much as saying the real number 2.000... is equal to the natural number 2 requires some mathematical sleight of hand, and the problem only gets worse the more exotic your number system gets.
For the purposes of size comparison, this isn't much of a problem - the number system is implicitly whatever the other person's using.
It might be prudent to give specific values of ∞ for different number systems, to get a feel for this. Specifically, this is what you would get if you embedded the number system in a stronger one, which is frought with the same ambiguities as raw ∞, but I digress. I'm only going to be giving examples for a small collection of well-behaved number systems, and you can generalise.
The infinity over the trivial number system 0, which is closed under multiplication and addition, is something like 1. No matter which number in 0 you pick, 1 is always larger than any of them.
The infinity over the reals, rationals and naturals is our friend, ω. ω is a number larger than any real, rational or natural number, and it's the simplest such number. The criterion for being simplest is arbitrary, but it can be justified since ω isn't actually a real, rational or natural so the whole mapping doesn't really make sense and I'm just doing this to illustrate to the reader. If I wanted to make it formal I'd say something like if the reals are embedded in a number system where ω behaves normally, then ω is the largest number that (real infinity) can not be proven to be greater than.
The infinity over the hyperreals, defined by having the real numbers as well as infinites and infinitesimals, is ω1. Kind of - the hyperreals represent a whole bunch of models, and some of them are much stronger or weaker than this. The standard one, though, uses only countable ordinals as their infinities and their multiplicative inverses as their infinitesimals, in the name of keeping the hyperreals within ℵ1.
Now we're getting into the big leagues. Over the set of all ordinal numbers, the infinity has its own special name - Cantor's absolute infinity. It's pretty easy to get bigger infinities, really - all you need is a number system stronger than the ordinals, so Cantor was hyping over nothing - but there it is.
I've decided to call the strongest number system ALL here. This might be a real absolute infinity - I've defined ALL such that you can't make a stronger number system, even just by making trivial extensions of ALL. This also leads to real contradictions, since it's defined over number systems in general, and you can't get out of it easily like Cantor's absolute infinity by declaring it a second ordinal.
Note that phrases like "bigger than ∞" don't work unless you've somehow managed to beat ALL - if you really have a number system that has numbers bigger than ∞, then there's also a new ∞ for the stronger number system that you're using, which is the important ∞ that you should be thinking about.