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)