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A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.

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What is the geometric description of the set of isomorphism class of $G$-torsors over a site...

You should read Section 4.5 of Olsson's book Algebraic Spaces and Stacks. The notion of a site is a piece of category theory with no intrinsic geometry, so it doesn't really make sense to ask for a …
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