Let $\mathcal{C}$ be a category and $A,B$ two objects in $\mathcal{C}$. We call $(P,\mu : P\rightarrow A,\nu:P\rightarrow B)$ a product of $A$ and $B$ if for any $P'$, $\mu':P'\rightarrow A$ and $\nu':P'\rightarrow B$ there exists a unique morphism $g:P\rightarrow P'$ such that $g\circ \mu'=\mu$ and $g\circ \nu'=\nu$.
Now assume that $F$ is a functor from the category $\mathcal{C}$ to category $\mathcal{D}$. What are the necessary conditions that whenever $F$ satisfies them, it sends products to products, namely, if $(P,\mu,\nu)=A\times B$ then $(F(P),F(\mu),F(\nu))=F(A)\times F(B)$?
Clearly, there must be some conditions since for any category $\mathcal{C}$ you can define a functor $F$ that takes every object to the set $X=\{0,1\}$ and every morphism to $id_X$ in the category Sets. Then $F(A\times B) = (\{0,1\},id_X,id_X)$ which is clearly not the product of $F(A)=\{0,1\}$ and $F(B)=\{0,1\}$ (in Sets).
However, if $(F,G)$ is an equivalence of categories I think $F$ takes products to products.
I am pretty new to the subject, so simple examples and explanations would be very nice.
You are right that any equivalence of categories preserve products; in fact it is enough that $F$ is a right adjoints, and then it preserves all limits (and in particular, products).
For example, the forgetful functor $\Bbb{R}-\mathrm{Vect}\to \mathrm{Sets}$ sends products to products; indeed, the product of two vector spaces (with componentwise operations) satisfies the universal property defining products in the category of vector spaces. Another example is the forgetful functor $\mathrm{Ab}\to \mathrm{Grp}$; indeed, the product of two abelian groups is an abelian groups, and satisfies the universal propriety in $\mathrm{Grp}$ (and thus also in $\mathrm{Ab}$).
Being a right adjoint is not a necessary condition to preserve products, however. For example, the abelianisation functor $()_{ab}: \mathrm{Grp}\to \mathrm{Ab}:G\to (G)_{ab}=\frac{G}{[G,G]}$ preserve products but it is not a right adjoint. But a functor from a complete category that preserves all limits and satisfies an additional condition is a right adjoint; this is a theorem known as Freyd's Adjoint Functor Theorem (more info here, or look at Chapter 5 of "Categories for the Working Mathematician" by Saunders Mac Lane).