When they say $\text{Hom}(A,B) \approx \text{Hom}(C,D)$ in category theory, what do they mean? For instance in Weibel.  Do they mean that the two hom sets are bijective or something in addition to that?

Show that $\text{Hom}_{\mathcal{C}}(A, \prod_i C_i) \approx \prod_i \text{Hom}_{\mathcal{C}}(A, C_i)$ and ...

No more info about general category $\mathcal{C}$ is given.
 A: This is a correspondence of sets. Assume the categories are locally small. 
First note that if $f:B\to A$, then there is the induced arrow 
$f^{*}:\text{Hom}_{\mathcal{C}}(A, \prod_i C_i)\to \text{Hom}_{\mathcal{C}}(B, \prod_i C_i)$ defined by $\phi \mapsto \phi \circ f.$
We also have the arrow
$\theta_f:\prod_i \text{Hom}_{\mathcal{C}}(A, C_i)\to \prod_i \text{Hom}_{\mathcal{C}}(B, C_i)$ defined by $(g_i)_{i\in I}\mapsto (g_i\circ f)_{i\in I}$
Then, $\approx $ means two things:
1). the correspondence is bijective: this is clear if we just note that by the UMP of the product, 
$\left (\phi: A\to \prod  C_i \right )\leftrightarrow (\pi_i\circ \phi )_{i\in I},\ $is a bijection, where $\pi_i$ are the canonical projections from the product. 
Call this bijection $()^{\star }$.
2). $()^{\star }$ is $\textit { natural in A }:\ $
if $f:B\to A$ then $(\phi \circ f)^{\star }=\theta_f((\phi )^{\star })$. 
The following calculation shows that this is true:
compute the LHS:
$(\phi \circ f)^{\star }=(\pi_i\circ \phi\circ f)_{i\in I}$
and the RHS:
$\theta_f((\phi )^{\star })=\theta_f(\pi_i\circ \phi)_{i\in I}=(\pi_i\circ \phi \circ f)_{i\in I}$
