Maxima is a computer algebra system, and can calculate using both symbolic (exact) and numerical expressions. Maxima can perform mathematical operations involving polynomials, trigonometric and exponential functions, differentiation, integration, Taylor series, differential equations, linear algebra, number theory, and more. Maxima can plot functions and data in two and three dimensions, and also includes basic programming capabilities. wxMaxima is a graphical user interface for Maxima that produces nicely formatted output; wxMaxima is included in most Maxima downloads and is the recommended way to work with Maxima.

While Maxima and Wolfram|Alpha have many of the same capabilities (and both are free!), some key differences are:
(1.) Wolfram|Alpha is a web-based program while Maxima must be downloaded/installed
(2.) Wolfram|Alpha has less rigid syntax and can show the intermediate steps in a computation such as differentiation or integration.
(3.) Maxima supports multi-step computations, calculations can be saved, and the appearance of function graphs can be customized.




Maxima website: http://maxima.sourceforge.net/
Download at: http://maxima.sourceforge.net/download.html
Manual/Tutorials: http://maxima.sourceforge.net/documentation.html
Portable version (can run from your USB drive): http://portableapps.com/node/11261

I have written an example-based tutorial for Maxima that can be viewed online at this link or downloaded from the link at the bottom of the page, along with the corresponding wxm (wxMaxima worksheet) file.