Q:

Can a classifying space be characterised universally?

I'm having trouble understanding what classifying spaces are in general.
It seems to me, that they are terminal in a category of bundles whose morphisms are pullback squares, is this correct?

A:

Actually a classifying space $BG$ for the group $G$ is weakly terminal (existence but not uniqueness) in the category of principal $G$-bundles with commuting squares for maps. Sometimes this sort of thing is called 'generic' in category theory, in this case the generic $G$-torsor. Also, depending on which model you take, it may not classify all bundles, for example Segal's model for the classifying space of a well-pointed topological group only classifies numerable bundles: those trivialisable over an open cover with a subordinate partition of unity. Off the top of my head I can't think of a universal bundle that classifies all principal bundles (such a thing would have to not be numerable itself, as Segal's and Milnor's constructions are). 
Note that given a collection of transition maps $U_{ij} \to G$ over a non-numerable cover $U = \lbrace U_i \rbrace$ of some space $X$ ($G$ well-pointed, here), there is another reason you cannot get a classifying map $X\to BG$, namely that the canonical map $|NU| \to X$, where $|NU|$ is the geometric realisation of the nerve of the topological groupoid associated to $U$, is not split, as it is in the case when $U$ is numerable, though it is a weak homotopy equivalence. This raises the question of whether the functor $X \mapsto GBund_X$ is representable on the homotopy category of $Top$, rather than on the category of spaces and homotopy classes of maps. This depends on whether or not an arbitrary space is weakly equivalent to a paracompact space.

