Berci has pointed out in a comment that, in order to talk about open balls, one needs a metric; there is no notion of ball in a general topological space. It might also be worth pointing out that, in separable metric spaces, one has the stronger property that there is a countable family of open balls (namely those with rational radii and centers in a given countable dense set) such that every open set is a union of a subfamily of this particular family. One could try to generalize this last observation by asking whether, in a separable topological space, there must be a countable family $\mathcal B$ of open sets such that every open set is the union of a subfamily of $\mathcal B$. This property of a space is called the second axiom of countability (and such spaces are called second-countable), and it is known not to follow from separability. Perhaps the nicest counterexample is the product (with the usual product topology) of $\kappa$ copies of a 2-element discrete space. This is separable for all cardinal numbers $\kappa$ up to and including the cardinality of the continuum (and not for any larger $\kappa$), but it is second-countable only for countable $\kappa$.
Some tangential comments: Instead of generalizing the observation that I began with, one could similarly try to generalize the original, weaker statement: Must there be a family $\mathcal B$ of open sets such that every open set is a union of a subfamily? Unfortunately, this generalization is silly; the answer is trivially affirmative (whether or not the space is separable) as we can just take $\mathcal B$ to contain all the open sets.
Although I described what I consider the "nicest" counterexample above, I must admit that my personal favorite is another counterexample: the Stone-Cech compactification of a countably infinite discrete space.
Finally, I emphasize that, although I'm now a senior citizen, terrible terminology like "first-" and "second-countable", "first" and "second category", was around long before I learned it. I officially disclaim any responsibility.