All fields are division rings. A more typical example is given by the quaternions. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring. In general, if R is a ring and S is a simple module over R, then the endomorphism ring of S is a division ring; every division ring arises in this fashion from some simple module.
All of linear algebra may be formulated, and remains correct, for modules over division rings instead of vector spaces over fields. Every module over a division ring has a basis; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gauss-Jordan elimination algorithm remains applicable.
All finite division rings are commutative and therefore finite fields (Wedderburn's theorem).