The prototypical example of a spin chain is the Heisenberg model, described by Werner Heisenberg in 1928. This models a one-dimensional lattice of fixed particles with spin 1/2. A simple version (the antiferromagnetic XXX model) was solved, that is, the spectrum of the Hamiltonian of the Heisenberg model was determined, by Hans Bethe using the Bethe ansatz. 
Now the term Bethe ansatz is used generally to refer to many ansatzes used to solve exactly solvable problems in spin chain theory such as for the other variations of the Heisenberg model (XXZ, XYZ), and even in statistical lattice theory, such as for the six-vertex model. 
Another spin chain with physical applications is the Hubbard model, introduced by John Hubbard in 1963.
This model was shown to be exactly solvable by Elliott Lieb and Fa-Yueh Wu in 1968.
Another example of (a class of) spin chains is the Gaudin model, described and solved by Michel Gaudin in 1976