Consider two transitive groups on the same set: $G = \langle (1,2,3) \rangle$ and $H=\langle (1,2), (1,2,3) \rangle$ both acting on $\Omega = \{1,2,3\}$.
How well do $G$ and $H$ swirl the points of $\Omega$? Well they both do a very thorough job! They are both transitive groups. If the boss wants 1 moved to 3, no problem, $(1,2,3)^2$ will do the job in both $G$ and $H$. Now move 3 back to 2? No problem! $(1,2,3)^2$ again will do the trick.
What if the boss goes mad with the power? What if he wants you to move 1 to 2 and at the same time leave 3 alone? In $G$ this cannot be done. Every element other than the identity moves all the points ($G$ is regular). In $H$ this is easy, $(1,2)$ does the job.
What is the difference between these groups? They are both transitive, but once you start saying where one point goes, they react very differently about what they can do to the rest.
In other words, $G_3=\{()\}$, but $H_3=\langle (1,2) \rangle$. The first group cannot do anything, but the second is transitive on $\Omega \setminus \{3\}$.
To understand a group action, it is not enough to just ask if it is transitive. We often want to know how the group can move pairs of points.
Consider the actions of $G$ and $H$ on $\Omega \times \Omega$, the set of ordered pairs $\{ (i,j) : 1 \leq i,j \leq 3 \}$. Clearly these actions are not transitive, because $(i,i)^g = (i^g, i^g)$. One orbit is always the "diagonal orbit" $\{ (i,i) : i \in \Omega \}$. What about the rest of $\Omega \times \Omega$?
Well for $G$ there are two orbits: $\{ (1,2), (2,3), (3,1) \}$ and $\{ (2,1), (3,2), (1,3) \}$. $G$ moves them in a circle, and does not change clockwise versus counter-clockwise. We say $G$ is a rank three permutation group.
For $H$ there is only one orbit: any $i,j$ with $i \neq j$ can be sent to any other such pair. We say $H$ is a 2-transitive permutation group.
For groups acting on graphs, we restrict what part of $\Omega \times \Omega$ is allowed and of concern, and "transitive" becomes "vertex transitive" while "2-transitive" becomes "edge-transitive".