Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3dimensional orbifolds? To be more specific, let Q be a 3dimensional orbifold. If the orbifold fundamental group of Q is a hyperbolic group (in Gromov sense), then can we say that Q is a hyperbolic orbifold?

[See Peter Scott's Bulletin article for more information.] Typically, we say an orbifold $Q$ is hyperbolic if it comes to us as a quotient of hyperbolic space $H^n$ by the action of a discrete group $G$ of isometries. If the action $G$ is cocompact then $G$ will be a Gromov hyperbolic group. This is the "easy direction". On the other hand, if $Q$ is an orbifold with enough topological hypotheses (for example, dimension three, irreducible, "good" as Agol says, perhaps more...) then, if the orbifold fundamental group of $Q$ is Gromov hyperbolic it follows from the geometrization theorem (Perelman and so on) that $Q$ is orbifold homeomorphic to a quotient as in the first paragraph. So, roughly, the two notions are equivalent. However one direction is easy  it follows from basic definitions in the field of coarse geometry  and the other direction is one of the most famous recent results in mathematics. 

