What does it mean that a category satisfies right Ore condition? What is the right Ore condition and what does it mean that a category satisfies right Ore condition?
Also can you please give an example of a category which satisfies this condition?
 A: It would be great if you could supply us with more context here. Are you interested in a category $\mathcal{C}$ satisfying the right Ore condition or rather a collection of morphisms $S\subset \text{Mor}(\mathcal{C})$ to satisfy the right Ore condition? These are somewhat different things.
Following nlab, a category $\mathcal{C}$ satisfies the right Ore condtion if for every two morphisms $A\to B$ and $C\to B$, there exists an object $D$ and morphisms $D\to A$ and $D\to C$ yielding a commutative square. Any abelian category satisfies this condition and $D$ is simply the pullback of $A\to B\leftarrow C$. More generally, any additive category having all kernels also has all pullbacks and thus satisfies the right Ore condition.
A slightly different animal is the following: Let $\mathcal{C}$ be a category and let $S\subseteq \text{Mor}(\mathcal{C})$ be a collection of morphisms. Then $S$ is said to satisfy the right Ore condition if for any morphism $A\to B$ and any morphism $s\colon C\to B$ with $s\in S$, there exists an object $D$ together with morphisms $D\to C$ and $t\colon D\to A$ such that $t\in S$ and $ABCD$ is a commutative square.
In fact, $S$ is called a right multiplicative system if it is closed under compositions, satisfies the right Ore condition as above and one more axiom (see the definition of a right multiplicative system here).
The point of a right multiplicative system $S$ in a category $\mathcal{C}$ is that you can build a new category $\mathcal{C}[S^{-1}]$ obtained by declaring all morphisms in $S$ to become isomorphisms. Thanks to the structure on $S$, the category $\mathcal{C}[S^{-1}]$ is relatively easy to understand.
A decent example of the latter is the following: Let $\mathcal{A}$ be an abelian subcategory and let $\mathcal{S}$ be a Serre subcategory, i.e. given a short exact sequence $$A\to B\to C,$$ $B\in \mathcal{S}$ if and only if $A,C\in \mathcal{S}$. Define a system $S\subseteq \text{Mor}(\mathcal{A})$ by those morphisms $s$ such that $\ker(s),\text{coker}(s)\in \mathcal{S}$. Then $S$ is both a left and right multiplicative system. In fact, the localized category $\mathcal{A}[S^{-1}]$ is again abelian.
