A cluster algebra, in its simplest form, is a subring of the field of quotients of polynomials in a finite number of indeterminates. The algebra is defined using initial data (a sign-skew-symmetric matrix and a free generating set of the field), known as a seed. Via a combinatorial process known as mutation, a subset of the field is produced; this is then a generating set for the cluster algebra.
Cluster algebras possess an interesting combinatorial structure. For example, those whose generating set above is finite are classified by the Dynkin diagrams. The cluster algebra corresponding to a Dynkin diagram is closely connected to the corresponding root system. The number of generators of such cluster algebras can be regarded as generalised Catalan numbers; indeed there is a polytope associated to each finite type cluster algebra which can be regarded as a generalisation of the classical associahedron, or Stasheff polytope, of binary trees.
Cluster algebras were introduced by Fomin and Zelevinsky [FZ02] in order to study the dual canonical basis of the quantum group of a simple Lie algebra over the complex numbers. The canonical basis was the culmination of much effort to understand the representation theory of simple Lie algebras, and describing and understanding it is still a big open problem in Lie theory. There are many connections between cluster algebras and other fields (including the representation theory of algebras and discrete integrable systems, Y-systems).
The course will be an introduction to cluster algebras from the beginning.
Some references to get started: