Is there a "universal group object"? (answered: yes!) I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$.  One problem with this idea is that this diagram $D$ as a category on its own doesn't have enough structure to make the object labelled $``G\times G"$ really the product of $G$ with itself in $D$.

Is there a category $U$ with a group object $G$ in it such that every group object in every other category $C$ is the image of $G$ under a product-preserving functor $F:U\rightarrow C$, unique up to natural isomorphism? 

(It's okay with me if "product-preserving" or "up to natural isomorphism" are replaced by some other appropriate qualifiers, like "limit preserving"...)
 A: Perhaps this is a good spot to put my standard propaganda that the definition of "group object" is wrong.  Recall that having inverses is typically a property not a structure.  The problem is that the inverse is typically not a morphism, instead it's an "anti-morphism."  This is true anywhere that the notion of anti-morphism makes sense (in particular: noncommutative rings, Poisson manifolds).
A: Yes, the category U is the opposite of the full subcategory of Grp on the free groups on 0, 1, 2, ... generators.  This is an instance of Lawvere's theory of "theories".  See this nLab entry for a discussion (of this example in fact).
A: I would really suggest looking at Steve Awodey's lecture notes on Categorical Logic., found here http://www.andrew.cmu.edu/user/awodey/catlog/notes/  The category you are looking for is called "the theory of groups".  I find these notes much more digestible than Lawvere's original papers on the subject. 
Essentially, you can form a category in which all objects are products of a single object G, and the only morphisms between them are those morphisms you get out of the basic definition of a group object.  Then a group in any category is a product preserving functor from this category.  Actually, in this case, it is easy to see that the appropriate category is just the opposite of the full subcategory 1, F(1), F(2), ... , F(n), ... where F(k) is the free group on k generators.
A: To elaborate on Reid's answer, there is a general notion of "operad" which provides universal "algebraic" objects in the sense that you want.
One interesting feature of groups: a group object is a product-preserving functor from a certain category.  A Hopf algebra object is a monoidal functor from a related category, but where the monoidal structure need not be product.  See, if your monoidal structure is Product, then every object of your category is a coassociative counital coalgebra in a unique way, and so half of the Hopf axioms are trivial.
Why does this matter?  There are no interesting group objects in Vect.  Indeed, there are no interesting (unital) monoid objects, because the product in Vect is also a coproduct.  A monoid object is an object V and maps e : {pt} → V and m : V x V → V, unital and associative.  Well, in Vect x is ⊕, and {pt} is 0.  There is only one linear map from 0 to V, so we know what e is.  If the monoid is unital, then m(v \oplus 0) = v = m(0 \oplus v), and linearity of m takes care of the rest.
