What does it mean to be a transitive permutation? Let $X$ be a finite set and let $G$ be a group. What is the meaning of $G$ being a "transitive permutation" on the set $X$?
 A: You say $X$ is a finite set, not a group.  Thus "on group $X$" (quoting your question as it now stands) is wrong.  One may say $G$ is a transitive permutation group on the set $X$, but not on the group $X$ unless $X$ is a group.
It makes sense to speak of a transitive permutation group on a set $X$, but one does not say "$G$ is a transitive permutation".  It is the permutation group that is transitive on the set $X$; the thing that is transitive is not a permutation.  Your subject line as it now stands asks about a "transitive permutation".  Again, it is not the permutation, but the group of permutations, that is transitive on the set $X$.
Parse the phrase like this:

$G$ is a transitive $\left\{\text{permutation group}\right\}$ on $X$,

not like this:

$G$ is a $\left\{\text{transitive permutation}\right\}$ group on $X$.

That $G$ is a transitive permutation group on $X$ means that for every pair $x,y\in X$, there is some permutation $g$ in the group $G$ that moves $x$ to $y$.
For example, the group of all shifts parallel to the $x$-axis in the $(x,y)$-plane is not transitive on the plane because there is no shift parallel to the $x$-axis that moves $(0,0)$ to $(1,1)$.  On the other hand, the group of all translations of the plane is transitive on the plane.
