Applications
The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups.
To make this theory explicit, there must be defined the concept of L-homomorphism of an L-group into another. That is, L-groups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but L-homomorphisms must be 'over' the Weil group.
Read more about this topic: Langlands Dual