• COIN-OR: COmputational INfrastructure for Operations Research. Includes optimization software, heuristics, and more.
• Gnumeric: a spreadsheet with built-in solvers GLPK and lp_solve.
• GLPK: linear and integer programming, including a parser for a subset of AMPL.
• GOBLIN: Graph Object Library for Network Programming Problems. Includes a C++ library for graph optimization as well as a graph browser and editor.
• OpenForecast: I have not used this, but it looks promising.
• FLOPC++: An algebraic modelling language implemented as a C++ class library.
• Zimpl: language to translate the mathematical model of a problem into a linear or (mixed-) integer mathematical program expressed in .lp or .mps file format.
• Cliquer: routines for clique searching.

Mathematical/statistical applications

• R: statistics, graphics, and more. Similar to S-plus (both are based on the language S).
• Maxima: computer algebra, similar to Mathematica or Maple.
• Octave: matrix based mathematics, similar to and "mostly compatible" with MATLAB.
• GNU Scientific Library: C library for mathematical functions, including random variables, statistics, linear algebra, and lots more.
• PSPP: similar to SPSS. It is not too far along in its development, however.

Manufacturing

• Qcad: 2-dimensional CAD. Only the X11 version is open-source and even then only when released as source code (so you have to compile it yourself). If you are in Pitt's School of Engineering and are interested in trying this product, I am willing to help compile it from source so you don't have to license the binary version. On the other hand, paying a small fee for the convenience of binaries may be worthwhile if you want to support the project.
• OpenCascade: 3-dimensional modeling.
• OpenFEM: finite element analysis.
• iCivilEngineer list of open-source software: includes additional CAD, finite element analysis, and more.