What is Matlab and when should I use it?

Matlab is designed for vector- and matrix-based numerical calculations. If you are doing anything that requires numerically solving matrix equations, it is probably the most convenient option. Because Matlab makes it very easy to visualize data quickly and with minimal fiddling with data, I use it for running monte carlo simulations, numerically solving ODEs, PDEs, integro-DEs, image processing and data analysis, generating figures for papers, lectures, and notes and many other purposes. It does take a bit of time to figure out how to use it, though. Note that Octave is a free software package that is mostly compatible with Matlab.