Engauge Digitizer

http://digitizer.sourceforge.net/
If you only have an image of a graph from scientific journal and you want to know the numerical values of curves/points on the graph, then you need to digitize the graph. This is a reverse process in comparison to plotting process. This open source, digitizing software can convert an image file of a graph or map into numerical data.

GNU Data Language (GDL)

http://gnudatalanguage.sourceforge.net/
If you use IDL (Interactive Data Language) in your research then a free program GDL can run your IDL code. The current version of GDL is compatible with IDL 6.0.

Texmaker and Lyx

If you need to write a high quality document for publication, then using LaTeX is a must. To simplify writing a latex document, I usually use either Texmaker or Lyx.

http://www.xm1math.net/texmaker/
Texmaker is a free LaTeX editor which has many tools needed to simplify writing LaTeX documents. We can use Texmaker on unix, macosx and windows systems.

http://www.lyx.org/
LyX is a document processor that combines TeX/LaTeX with a graphical interface. It can be used create mathematical content (using a fully integrated equation editor) and structured documents such as articles, theses, and books.

Plotting using Gri and Gnuplot

http://gri.sourceforge.net/
Gri is command-driven tool for creating scientific plots. Gri can create x-y graphs, contour graphs, and image graphs.

http://www.gnuplot.info/
Gnuplot is a copyrighted but freely distributed, portable command-line driven plotting tool. Gnuplot can make many types of 2D and 3D plots.

Numpy and Scipy

http://numpy.scipy.org/
Numpy is Python module for numerical computations. Numpy contains a N-dimensional array object, basic linear algebra functions, Fourier transforms, random number capabilities, and tools for Fortran and C/C++ code integration.

http://www.scipy.org/
SciPy is open-source Python module for scientific programming. It uses NumPy to provide N-dimensional array manipulations. "If you need to manipulate numbers on a computer and display or publish the results, give SciPy a try!"

Python and VPython

http://www.python.org/
Python is a free, open source, multi-platform, object-oriented programming language created by Guido van Rossum in 1990. With Numpy module, Python is powerful to do numerical calculations such as matrix manipulations.

http://www.vpython.org/
VPython is a "visual" module for python language. We can create easily three dimensional objects such as spheres, curves, and cubes. We use this module to do simulations.

Maxima

http://maxima.sourceforge.net/
Maxima is a computer algebra system, similar to Mathematica. It can do symbolic and numerical manipulations. It can also do two and three dimensional plots.
There is a cross platform GUI for the maxima based on wxWidgets called wxMaxima. See:
http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page

GNU Octave

http://www.gnu.org/software/octave/
GNU Octave is a mathematical software which is mostly compatible with Matlab and mainly intended for numerical computations.

Scilab

http://www.scilab.org/
Scilab is a mathematical software for numerical computation which is similar to Matlab. This is very useful for physical computations. It is developed by INRIA and École nationale des ponts et chaussées (ENPC). It can be downloaded for free at the above links. See also the license. The syntax of scilab is similar but not compatible to Matlab. However we can convert Matlab format to Scilab format easily using a converter included in the Scilab. There is also a nice package called Scicos for modeling and simulation. I use Scilab mainly for plotting, data processing, numerical calculations and solving matrix equations.