Permutation group equivalence relation, does this equivalence have a name?

Question

Given any permutation group $P$ on a set $S$ so that $P\subseteq \text{Sym}(S)$ one can define an equivalence relation $\sim$ on $S$ so that we have:

$$\forall x,y\in S\left(x\sim y\iff \exists \sigma\in P:\sigma(x)=y\right)$$

Thus intuitively two elements are equivalent, if some permutation in $P$ is capable of swapping them. Does this equivalence relation have a name? Is it of any use?

I know very little group theory, but basically I've been trying to understand the ways lets call them "items" appear different when viewed from different perspectives. Now two items may appear different but it might just be I'm viewing them differently and in reality they are the same. The permutation basically accounts for the different perspectives. In this way the equivalence relation tells me to some extent how far/or how capable which parts of my item can appear by changing my perspective of the item.

Answer 1

The group $P$ acts on the set $S$, and the equivalence classes are called the orbits of the action of $P$ on $S$. See also wikipedia for more related terminology.

Answer 2

The relation forms an partition of the set $S$, each subset that is in the partition is called an orbit.
We normally just don't have such terminology for the relation ~ itself though. Your problem basically resembles the coloring problem. And generally if you want to count the number of orbits you may want to apply the Burnside's Lemma or Polya's Enumeration Theorem.
Think of it this way, you got an object with some sort of symmetry within, for example, an square that remains the same under rotation and reflection about some axis. If you paint colors on the vertices, eventually there're some seemingly different ways to paint that is the same under the symmetry of the square.