When do I use "arbitrary" and/or "fixed" in a proof?

Both "arbitrary" and "fixed" are just shorthand for a universal quantifier. When I say something like "fix $\epsilon > 0$" it means I am about to prove a statement that is true for all $\epsilon > 0$ (and thereby prove that some function is continuous, for example) but I don't want to actually write out "for all $\epsilon > 0$" in front of every sentence I'm about to write. That's really all there is to it.

When you encounter the term "arbitrary", it usually just means that a given statement is specified for any element from a given set of elments. For example, if I say, let $x$ be an arbitrary element of the interval $[0, 1]$ I just mean that $x$ can take on any value within that interval.

The term fixed connotes a similar but more specific meaning. If I say let $x$ be a fixed element of the interval $[0, 1]$ I mean that, firstly, $x$ is an arbitrary element of $[0,1]$ but it's value is unchanging througout its usage.

The difference in meaning is subtle but sometimes important. If you start out with a "fixed" element you can't, for example, choose its value at some later point in the argument. On the other hand, if I prove that something holds for an arbitrary element of a given class, then it will hold for any particular element of that class which can often be very useful.