I've studied Root Systems to classify Lie Algebras and the construction emerged pretty naturally. Anyway I've been said that Root Systems are applied in other fields and are a useful tool in classify other algebric structure.
If anybody knows something about I'd like to have some hint on both where Root Systems apply and if there some kind of general idea of why they're so useful.
Root systems are one of the most remarkable structures elucidated in 20th century mathematics. They have a simple definition in terms of linear algebra and combinatorics, but have very far reaching applications in many areas of mathematics and physics. Most notably root systems appear in the classification of Lie groups and Lie algebras, algebraic groups, in representation theory of Lie algebras, Reflection Groups and Coxeter Groups, in hypergeometric functions, Macdonald conjectures, Selberg integrals, in the structure theory of reductive groups, and in the theory of symmetric spaces, just to name a few. For references see also here.