What is the difference between "order" and "cardinality"? I'll ask my question in the form of an example:
Consider the fundamental group $\pi_1(X)$ of a topological space.
The elements of this group are homotopy (i.e., equivalence) classes of loops in $X$.
Is it equally correct to refer to the "order" and the "cardinality" of each equivalence class in the group?
I often get this mixed up, since "order" can also refer to the smallest power of a group element which produces the identity.
However, in this case, the order of an element of $\pi_1(X)$ would refer to the number of loops that are homotopy equivalent to each other, yes?
In most cases I have seen only "order" used to refer to groups, and "cardinality" to refer to sets, but if an equivalence class is really a set, is there any reason not to refer to the cardinality of it?
Is this all preference, or is there a certain convention to these things?
 A: 
In most cases I have seen only "order" used to refer to groups, and "cardinality" to refer to sets.

Correct. For historical reasons, we call the cardinality of the underlying set of a group the "order" of the group. Also, as you mentioned, the order of an element $g$ of a group is the least positive integer $n$ such that $g^n$ is the identity element.
You should not use "order" in place of "cardinality" in almost any other context! We really shouldn't even use it for groups, but we're bound by tradition.
So:

Is it equally correct to refer to the "order" and the "cardinality" of each equivalence class in the group?

If you talk about the "order" of an element of $\pi_1(X)$, this can only mean its order as an element of the group $\pi_1(X)$, i.e. the least positive integer such that blah blah blah. If you talk about the "cardinality" of an element of $\pi_1(X)$, this can only mean the number of loops in the given equivalence class. These are of course very different things, so make sure to say the one you mean!
