Here is the definition of principal $G$-bundle in "differential geometry" written by Taubes.
Fix a smooth manifold $M$, and a Lie group $G$. A principal $G$-bundle is a smooth manifold $P$, with the following extra data: (1) a smooth action of $G$ by diffeomorphisms; thus a map $m:G\times P\to P$ with the property that $m(1,p)=p$ and $m(h,m(gp))=m(hg,p)$. It is customary to write this action as $(p,g)\mapsto pg^{-1}$.
I'm quite confused that why we use the notation $pg^{-1}$, since when I refer to other materials, they always use $p\mapsto gp$. So why does Taubes use $pg^{-1}$ here?
To clarify the comments above:
Given a right action of a group $G$ on an object/set $X$ one can always obtain a left action by the formula: $$ g\cdot x = xg^{-1}.$$
The point of the inverse is to conserve the multiplicative structure - i.e., to have a homomorphism from $G$ into the group of automorphisms of $X$:
$$(gh) \cdot x = x (gh)^{-1} = x (h^{-1}g^{-1}) = (x h^{-1})g^{-1}= g\cdot (xh^{-1}) = g\cdot (h \cdot x).$$
Namely, eliminating the middle steps, one gets, $$(gh) \cdot x = g\cdot (h \cdot x),$$ as desired.
Of course, one can obtain a right action from a left action in the same manner.
[On the other hand, I don't know why the author wants a left action if he is starting with a right action on $P$ - and perhaps that was your question.]