Co-Objects are better This is a rather vague question, but perhaps we can talk about it.
There are two types of mathematical objects (which don't exclude each other):
A) There is a good description of morphisms defined on this object.
B) There is a good description of morphisms defined into this object.
Thus A) means that the covariant hom-functor is understood, and B) means that the contravariant hom-functor is understood. This applies most notably to universal objects. Within category theory, the concepts are just dual to each other and so the "theory" of A) is essentially the same as the theory of B). But most categories studied in mathematics don't come together with their dual, so that this categorical argument is not really good. In fact, I have the feeling that in 'daily mathematics', A) appears much more often than B). And that it is easier to work with them. Of course, we could argue about that. For example, I have a better feeling with colimits than with limits. [perhaps I will add examples here]
If you have the same feeling: Can we give reasons for this?
I think that the basic principle of gluing, which appears in many geometric categories, always belongs to A). This could be a reason. What do you think?
 A: Dear Martin,
As Harry points out in his comments, in certain settings (e.g. moduli spaces) an object is characterized by maps in.  In others (e.g. the free abelian group on one generator), the characterization is by maps out.  
Certainly in algebra, injective objects (characterized by maps in) are typically regarded
as more mysterious and black-box like than projectives (where one can typically think of
free modules, which are quite concrete).  I know several situations in which someone made real progress by judicious use of injective objects, and I'm sure part of the obstruction to previous researchers was just that injectives are not as familiar; in short, there
is probably arbitrage to be gained for some (myself, at least) by learning more about injectives in various contexts,
and trying to use them as fluently as one uses free objects.  
In topology and geometry, perhaps there is more fluidity between the two characterizations. 
E.g. maps into the circle make it the Eilenberg-Maclane space $K({\mathbb Z},1)$, while
maps out define the fundamental group.
You are correct that quotienting by an equivalence relation (gluing) is related to
maps out.  Perhaps this is one reason why the construction of moduli spaces (e.g. Picard
schemes) can be quite involved; they are characterized by maps in, but are often
constructed by a gluing procedure, which creates conflict; thus one finds oneself working locally, and is led into sheaf/stack-theoretic issues.  
Certainly, the tension between the two characterizations has been a fertile source for
good mathematics.
