A topological space $(X,\tau)$ is connected if and only if the only continuous maps $f:X\to\{0,1\}$ (where $\{0,1\}$ carries the discrete topology) are the constant maps.
Are there other examples of topological properties that can be discribed via topological properties of continuous maps? (I apologize for the somewhat fuzzy nature of this question.)
There are numerous examples and it's not too difficult to come up with a few on your own, but here is a few examples:
A topological space $X$ is said to be functionally Hausdorff if and only if each pair of distinct points of $X$ can be separated by a continuous function $f:X\rightarrow [0,1].$
Another would be to say that a topological space $X$ is $T_0$ if and only if each initial source at $X$ is point separating.
A space $X$ is totally disconnected if and only if each continuous map from a connected space into $X$ is constant.
A space $X$ is connected if and only if every continuous map from $X$ into any totally disconnected space is constant.
In a way this is precisely what the theory of classifying spaces is about. For example the cohomology group $H^1(X)$ corresponds to homotopy equivalence classes of continuous maps from $X$ to $K(\pi,1)$ where $\pi$ is the fundamental group.
There are a few examples based on the lifting property in the category of topological spaces. For example, see answers there https://math.stackexchange.com/questions/434312/are-there-useful-categorical-characterisations-of-the-topological-separation-axioms
But there are also other examples, see http://arxiv.org/pdf/1408.6710.pdf or more examples in an unfinished update.
