Let $H⊂G$ be a subgroup and $π:Q→B$ be a principal $H$-bundle.
G has a left H action and one can define a principal G-bundle $π′:Q×_{H}G→B$ where $Q×_{H}G$ is quotiening out the diagonal H-action of $Q×G$. Morgan, Gauge Theory and the Topology of Four-Manifolds Define Reduction by A reduction of a $G$-bundle $P \to B$ to an $H$-bundle is a pair: an $H$-bundle $Q \to B$ and an isomorphism of $G$-bundles $ Q ×_{H} G \to P$.In Lectures on K¨ahler Geometry by ANDREI MOROIANU, define a reduction by If $H \subset G$ is a subgroup of $G$, a reduction of the structure group of $P$ to $H$ relative to the inclusion of $H$ in $G$ is a subset $Q$ of $P$ such that $Q \to M$ is an $H$-principal bundle over M.
I do not understand and relate these two definitions.
In my opinion, the simplest definition of a reduction of a principal $G$-bundle $P\to M$ to a subgroup $H\subset G$ is a principal $H$-bundle $Q\to M$ together with a smooth map $\Phi:Q\to P$ which covers the identity on $M$ and is $H$-equivariant. (The last condition makes sense, since $h$ is a subgroup of $G$, so you can require that $\Phi(q\cdot h)=\Phi(q)\cdot h$ for all $q\in Q$ and $h\in H$.)
Since the principal right action is free, and $\Phi$ covers the identity, it is a smooth embedding. Hence you can equivalently view the reduction as begin given by its image $\Phi(Q)\subset P$. This gives the definition Moroianu uses, although I would prefer to phrase it as "a subset $Q\subset P$ which is made into a principal $H$-bundle by the restriction of the principal $G$-action to $H$. (This condition can be phrased equivalently as the fact that $H$ acts freely and transitively on the intersection of $Q$ with each fiber of $P$ and that locally there are smooth sections of $P$ which have values in $Q$.)
On the other hand, given the map $\Phi$ from above, consider the map $Q\times G\to P$ defined by $(u,g)\mapsto \Phi(u)\cdot g$. By equivariancy of $\Phi$, this factors to a homomorphism $Q\times_H G\to P$ of principal $G$-bundles, which covers the identity on $M$ and hence is an isomorphism of principal bundles. This recovers the definition used by Morgan.
To return from Morgan's definition to the others, you just have to observe that there is a natural smooth map $Q\to Q\times_H G$ which covers the identity on $M$ and is $H$-equivariant. This is simply given by mapping $u\in Q$ to the class of $(u,e)$, where $e\in H$ is the neutral element. Combining this with an isomorphism $Q\times_H G\to P$ you get back to the other concepts.