I don't fully understand the definition of an Orbit. Mathematically, it is given by
$$
\operatorname{Orb}(x) = \{y = gx \mid g \in G\}
$$
where $G$ is a group and $x \in X$, a set that is acted upon by the group $G$, but what does this actually mean? Is it the set of all elements $x$ after having been acted upon by some element $g$?
The stabilizer is given by $\{g \in G \mid gx = x\}$. So does this mean the set of all elements $g$, when after actings upon $x$ give you the element $x$? That is, the set of all elements $g$ whose action on $x$ doesn't change it?
Best Answer
The orbit of $x$ is "everything that can be reached from $x$ by an action of something in $G$."
The stabilizer of $x$ is "the set of all elements of $G$ which don't move $x$ when they act on $x$".
Those already seem pretty intuitive... what else can be said?
I guess you might want to look at orbits and stabilizers for particular actions. For example, if a group is acting on itself by conjugation, then the orbit of an element is that element's conjugacy class. One element stabilizes another in this action exactly when they commute.