What is the algebraic interpretation of a contracted product?

Suppose you have a linear reductive group $G$ acting on an algebraic variety $X$. Let $Pleq G$ be an algebraic subgroup and let $Y subseteq X$ be a closed subvariety, invariant under $P$. Then a number of texts define the construction of the so called "contracted" or "twisted" product give by $G times^P Y := G times Y /sim quad quad$ where $(gp, y) sim (g, py)$ for all $p in P$.

My question is this: What is the algebraic interpretation of this geometric space and how should I be thinking about this scheme? In the case that it is affine, is there simple description of its coordinate ring?

Also does anyone have any references for a good exposition of these concepts? If there were a more categorical description of this object that would also be amazing (i.e. a definition given by diagrams etc.).

Thanks! Greatly Appreciated!