A (real) reflection is an automorphism of a real vector space that acts as negation on a one-dimensional subspace while fixing a hyperplane pointwise. The subspace is described by a vector called the root, while the hyperplane is described as the kernel of an element of the dual space called the coroot.
A root system is a collection of root/coroot pairs that is closed under the action of the corresponding reflections. Only finite root systems are supported at the present time. A root system gives a much more detailed description of a reflection representation of a finite Coxeter group.
Root systems are used to classify the semisimple Lie algebras. The closely related concept of a root datum is used to classify the groups of Lie type.
This is described in Chapters ROOT SYSTEMS and ROOT DATA.