What does it mean exactly by a metric "generates" a topology? For example, the discrete metric $d(x,y)$ where $d(x,y) = 1$ if $x\neq y$, $d(x,y) = 0$ if $x = y$ "generates" the discete topology $\tau$ where $\tau = 2^X$
Can someone clarify exactly what is meant by "generate"?
 A: In your basic calculus class, you define the notion of a continuous function using absolute values and $\epsilon-\delta$ definitions. The "distance" you are using is $d(x,y)=|x-y|$. 
It turns out, you can define continuity usefully on any metric space, using the distance function rather than $|x-y|$.
The truly surprising thing though, is that continuity of the function turns out to be less about the metric(s), and more about what sets are "open." Two different-seeming metrics can yield the same notion of which functions are continuous.
That notion of "open sets" arising from a metric, is what we call a topology on a set.
Now, it turns out, this notion does not require metrics at all. We can define a "topology" on a set $X$ as a set $\tau$ of subsets satisfying certain properties. We call those sets "open" in the topology - that is, $U\subseteq X$ is a open in $\tau$ just means that $U\in \tau$.
Having a topology on the domain and range of a function turns out to be enough to define continuity, and different topologies can give different types of continuity. 
So what we have is that metrics let you define topologies, but some topologies do not come from metrics, and different metrics on the same set can yield the same topology on that set.
(There are certain types of continuity that cannot be talked about in general topology - that is, without a metric. The primary one is "uniform continuity." For uniform continuity, you need a metric topology. or some symmetries on your space.)
A: I would say that the verb "defines," or perhaps "induces," provides a more accurate description of what is intended here by "generates." Using the metric, we can define the notion of an open ball, and then verify that these balls satisfy the axioms to be a basis for a unique topology on the set $X$ (the set on which the metric is defined). Really, with the usual use of "generate," it's the open balls defined by the metric which generate the topology (in the sense that the smallest topology on $X$ which contains all the open balls is the metric topology defined or induced by the metric). 
But to return to your original question, "generates" is being used here just to mean that the metric defines a topology in a canonical way. It's more of a colloquial use than technical (and I would prefer one of the two verbs I mentioned above). However, when we say that the open balls defined using the metric generate a topology on $X$, we are using the verb "generate" in a precise technical manner. 
A: The metric defines sets which we call "open": a set in a metric space is said to be "open" if every point has an open ball around it contained in the set.
Those "open" sets form also a topology. The collection of those "open" sets is the topology that the metric generates.
A: If you have a metric, you can define closed and open balls;
$$V = \{x: d(x,0) < r\}$$
(open balls)
Then say that the open sets are those that can be obtained by any union and finite intersection of open balls, and you have open sets. If you have open sets you have a topology 
