Compute Nash Equilibrium for 2-player games
We have developed a user-friendly browser-based system to input and solve 2-player strategic-form and extensive-form games:

http://www.gametheoryexplorer.org/ (GTE)

Currently, we support:


finding all equilibria via polyhedral vertex enumeration (which works on the strategic-form representation, which is converted to in the case of extensive-form games; this uses David Avis' lrs http://cgm.cs.mcgill.ca/~avis/C/lrs.htmlW); and
finding one equilibrium using Lemke's algorithm (which works on the strategic-form representation or on the sequence-form representation for extensive-form games).


You can use lrs separately offline to solve larger bimatrix games.

The GTE project is under active development (supported by the Google Summer of Code in 2011, 2012, and 2014), so please provide feedback and/or let us know if you would like to contribute) at

gte@nash.lse.ac.uk

For a paper that covers the basic theory and a number of methods for enumeration of equilibria, see 

D. Avis, G. Rosenberg, R. Savani , and B. von Stengel (2010).
Enumeration of Nash Equilibria for Two-Player Games.
Economic Theory 42, 9-37.

For a paper that describes GTE and what it can do, see

R. Savani and B. von Stengel (2014).
Game Theory Explorer - Software for the Applied Game Theorist.
Computational Management Science, 29 pages, to appear. arXiv version

Zouzias' answer gives a good description of how to use support enumeration, which works fine for small examples. The methods in the paper above and used in GTE will in general be much quicker than support enumeration for finding extreme equilibria. In addition, GTE will find the complete set of equilibria (which can include convex combinations of extreme equilibria) by finding maximal cliques in a bipartite graph.