Gluing together mathematical structures, how? By structure, I mean that which is defined here:
http://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29
What I'm looking for is a way of gluing together structures so that each structure used is embedded within the whole glued-together object.  (Each--meaning not just "most")
I don't need this embedding to be elementary; just something that "preserves" function, relation, and constant symbols.
I would think that some type of product would work.
If we have a set of structures $S$, then I want to show there exists a structure $U$ with the property that there is an injective homomorphism from every structure $A$ in $S$ into $U$.
The following link will help solidify what I mean:
http://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29#Homomorphisms_and_embeddings
Are there two cases, depending on the size of $S$, finite and infinite?   What I'm hoping for is some type of product of the structures in $S$ where $S$ is going to be a "large" collection of structures.
 A: Assuming you have a fixed signature (all structures in $S$ are algebras of the same type), there isn't always such an object.
For example, if you are working with rings with unity, it is not the case that given any collection of rings with unity $\{R_i\}$ there is always a ring with unity that contains each $R_i$ as a subring (for this situation, we require that subrings of $R$ have the same unity as $R$). For instance, there is no ring with unity that contains the field with 2 elements and the field with 3 elements as subrings-with-unity, because that would require the multiplicative identity to have both order $2$ and order $3$. On the other hand, in the category of rings (with or without unity, and homomorphisms are not required to preserve the unity), then the direct product and the "coordinate embeddings" work.
More generally, if you are working in a category of algebras where:


*

*For any two objects $A$ and $B$, there is always at least one morphism from $A$ to $B$; and

*For any family $\{A_i\}_{i\in I}$ of objects there is a (categorical) product,


then the product of the family of structures will always work (assuming the Axiom of Choice): for given a family $\{A_i\}_{i\in I}$, if $(P,\pi_i)$ is the product, then for pair $(i,j)$ let $f_{ij}\colon A_i\to A_j$ be an arbitrary morphism if $i\neq j$, and let $f_{ij}=\mathrm{id}_{A_i}$ if $i=j$. Then the family $f_{i_0,j}\colon A_{i_0}\to A_j$ induces a homomoprhism into the product $\mathcal{F}_{i_0}\colon A_{i_0}\to P$ such that $f_{i_0,j}=\pi_j\circ \mathcal{F}_{i_0}$ for each $j$; in particular, $\mathrm{id}_{A_{i_0}} = \pi_{i_0}\circ \mathcal{F}_{i_0}$, so $\mathcal{F}_{i_0}$ must be one-to-one, giving the desired immersion.
There are other circumstances where such an object may exist (that is, the conditions above are probably not necessary). But it seems hard to characterize when they do and when they don't in the abstract. 
