Introduction
Kan fibrations (AKA Kan complexes) are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets.
For a general discussion about fibrations see the page here.
Fibration and Cofibration
Homotopy has the concept of:
 a fibration which has the lifting property.
 a cofibration which has the extension property extension is dual to lift.
Fibration 
Cofibration (Extension Property) 


Homotopy  Fibration 
Cofibration (see page here) 
Combinatorics 
Kan fibration (see page here) 
Kan extension (see page here) 
Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces.
(ncatlab).
Extension is dual to lift.
Fibrations and Cofibrations are used in model theory.