Regarding to the category of groups (Grp), the trivial group is a terminal object. To show this, we must to exhibit a single morphism from any group to this trivial group.
What is a trivial group? I assume it is any group with just one element, the identity. Am I right?
What is this morphism we are looking for each group?
The trivial group is indeed the group with only one element, which must be the identity (group axiom). The operation is fixed: just $e\ast e =e$ is the only thing we can do and is forced by the group axioms, $e^{-1}=e$ etc.
It's clear that for any group $G$ we only have one function that maps to $\{e\}$, the constant one, and it's clearly a homomorphism of groups, so a valid arrow in $\textbf{Grp}$.