I am familiar with the notion of products and limits in category theory, but somehow struggling to understand the term "a category with finite products" or "with finite limits".
Does it mean that there are a finite number of products or limits in a given category?
Any help will be greatly appreciated.
If $A_1,\dots, A_n$ (not necessarily distinct, and $n=0$ is allowed) are objects in a category $\mathcal C$ then an object $P$ equipped with morphisms $p_i:P\to A_i$ serves as product of the $A_i$ if for every object $B$ and every tuple $(f_1,\dots,f_n)$ of morphisms with $f_i:B\to A_i$ there is a unique morphims $f:B\to P$ such that $f_i=p_i\circ f$ for $i=1,\dots,n$.
A category $\mathcal C$ has finite products if for every finite family of objects a product exists.
Special cases.
In special case $n=0$ we need an object $P$ such that for every object $B$ exactly one arrow $B\to P$ exists. This in order to guarantee the mentioned uniqueness. This means that $P$ must be a terminal object and the other conditions are then vacuously satisfied. So any category that has finite products will have a terminal element.
In special case $n=1$ a product of $A_1$ equipped with identity serves as product of $A_1$.
It is evident that in case $n=1$ a product exists. Further it can be proved that the existence of binary products (case $n=2$) implies the existence of $n$-ary products for $n\geq2$.
This together makes clear that for having finite products it is sufficient if the following conditions are satisfied: