In the case of a finite dimensional representation of a compact Lie group, one picks a basis in which the action of the a maximal torus T is diagonal. The weight associated to a vector in this basis is the homomorphism
lambda: T-->T^1 : t_1^lambda_1 . t_2^lambda_2 ... . t_n^lambda_n
by which the maximal torus acts on the vector. The weight space of the group representation is the set of weights of the representation in the charcter lattice of the maximal torus.
A clear exposition of this material can be found in Pressley and Segal : Loop groups chapter 2.