Mathematica

Mathematica is a computer algebra system sold by Wolfram Research.

Danger to Science and Mathematics
Please see this PDF written by Jordi G. H. originally from everything2, and follow up by reading this blog article commenting on the PDF.

Alternative
These are alternative, fully open source, and free computer algebra systems.
 * Maxima and wxMaxima
 * Sage
 * SymPy and SciPy
 * Axiom, OpenAxiom, FriCAS

Some other related libraries
 * mpmath, GMP, or MPFR for arbitrary precision computation
 * gnuplot for highly configurable and fast plotting
 * GNU Octave and QtOctave (front end)