it is a well-known theorem, first demonstrated by cantor, that:
if A is a set, then |A| < |P(A)|, whre P(A) is the power set of A, or set of all subsets of A.
if |A| is finite, this is obvious, and not very interesting.
if |A| = |N|, where N is the natural numbers, this yields the surprising result, that there exist uncountable sets. so that's 2 kinds of infinite: countable, and uncountable.
but if one continues with P(A), where A is uncountable, then one gets a set is that is "even more uncountable". it is possible to carry this reasoning on indefinitely, each time winding up with a "bigger infinity than before".
but there are other ways of thinking of "infinity" besides just "something not finite in size". for example, one can imagine that the distant horizon, is all one single point: the "point at infinity". this one single point (which is strange because we can approach it from any direction) acts like a boundary for what seems an endless plane, and it makes a flat euclidean plane act like a sphere (or the real number line act like a circle).
in this strange geometry, the hyperbola xy = 1 suddenly becomes a closed curve: the points at infinity connect. and this is yet another kind of infinity, which has nothing to do with number, per se, but has to do with space.
and here, again, we aren't limited to a single choice: instead of wrapping the plane into a sphere, we might want to distinguish between ±∞, which gives us a different kind of geometry.
and it gets weirder, still: we might declare some large number R to be the "largest finite number" (computers actually do something akin to this, called "overflow error handling"). now infinity is looking a lot smaller, and has "absorbing properties" similar to 0 (and, unfortunately, plays havoc with our usual algebraic rules).
these aren't the only possibilities. in some sense, infinity represents a choice: we might mean different things by it, and each meaning we assign to ∞, has different consequences. some of these definitions "break" the structure we add ∞ to: if we call ∞ a real number, we lose some of the field structure (algebra) of R. sometimes adding ∞ to our structure enhances it: if we add infinite numbers in a certain way, we can adapt induction to cover finite AND infinite cases.
i hope this gives you a small idea of how infinity comes in different flavors, it's not enough just to say: "∞".
