Let $C$ be a category and consider the product category $C \times C$. There is a diagonal functor associating to each object $X$ of $C$ the pair $(X,X)$ as an object of $C \times C$. On the other hand, there may be a product functor $C \times C \rightarrow C$, associating to $(X,Y)$ a product $X \times Y$. For example, this is the case in $Grp$. Convince yourself that the product functor is right-adjoint to the diagonal functor. If there is a coproduct functor, verify that it is left-adjoint to the diagonal functor. (Aluffi, Algebra: Chapter $0$, Chapter 8, Exercise 1.22)
What does it mean here a diagonal functor, left-adjoint, right-adjoint, and what does the diagram look like and how to start this problem.
I recommend reading a book on Category Theory, such as Steve Awodey's book, for instance. Aluffi is excellent for seeing category theoretic language in its natural algebraic setting, but I find he can gloss over some concepts a bit quickly. Tom Leinster also has an excellent (and free!) book on category theory which covers adjoints. Mac Lane's "Categories for the Working Mathematician" is a standard reference, but I found it too difficult when I was first getting started - perhaps you will have a different experience, though.
As for your question:
If $F : C \to D$ and $G : D \to C$ are functors between categories $C$ and $D$, they are called an Adjoint Pair and we say $F$ is Left Adjoint to $G$ (resp. $G$ is Right Adjoint to $F$), written $F \dashv G$, iff
$$\text{Hom}_D(FX,Y) \cong \text{Hom}_C(X,GY)$$
(note $F$ is on the left side of the hom, and $G$ is on the right side).
An important motivating example is $- \times A \dashv (-)^A$, where the left is the cartesian product functor and the right is the set exponential functor. Here adjointness says
$$\text{Hom}(X \times A,Y) \cong \text{Hom}(X, Y^A)$$
In more pedestrian terms, this says there is a bijection between functions $f : X \times A \to Y$ and functions $\tilde{f} : X \to Y^A$. This bijection is given by partial application, if you are familiar with currying.
Now, with this definition in mind, to show $\Delta : C \to C \times C$ is left adjoint to $\times : C \times C \to C$, we have to show
$$ \text{Hom}_{C \times C}(\Delta X, (A,B)) \cong \text{Hom}_C(X, A \times B)$$
By the definition of $\Delta$, this is
$$ \text{Hom}_{C \times C}((X,X), (A,B)) \cong \text{Hom}_C(X, A \times B)$$
But an arrow from $(X,X) \to (A,B)$ is exactly a pair of arrows - one $X \to A$ and one $X \to B$ (cf. the definition of product category).
But the universal property of products says exactly that such pairs of arrows are in bijective correspondence with arrows $X \to A \times B$. As needed.
Can you modify this proof to show the coproduct functor is left adjoint to $\Delta$?
I hope this helps ^_^