Something that is important to note is that there is not just one "infinity", but rather many, many infinities (in fact infinitely many infinities).
We say a set (collection of objects) is infinite if there is no 1-1 map into a set $\{1, 2, \ldots, n\}$ (1-1 just means don't send any two things to the same place). So now we have a set that is not finite. An example of this would be the integers, the even integers, the real numbers (ones you would experience in calculus), the rational numbers, complex numbers, and the list goes on. Now we arrive at the question of whether these infinities are the same.
In math, we say two sets $S$ and $K$ are the same size if there is a bijective map between the two, i.e. if we can assign to every $s$, a value $k$, such that every $s$ has a buddy $k$. To illustrate this, let's look at the even integers vs. all the integers. Are they the same size?
Well, let's define a map $f : \mathbb{Z} \to 2\mathbb{Z}$, sending $a \mapsto 2a$. This will assign to every integer an even integer, and every integer is "hit" by this map, so we can say that the two sets have the same size.
Showing two sets don't have the same size is a little more difficult. To see that the real numbers are bigger than the integers you can look up "Cantor Diagonalization".
You can also look up "cardinals and ordinals" to find information on different set sizes.