Inkscape is open-source vector graphics software, used widely and supported online by its users. Download here.
Here is a detailed tutorial, starting from the basics, by Mark Crutch.
Below are the steps for converting a bitmap picture (for instance of a drawing) into a vector format in Inkscape.
A bit of GIMP for processing a photograph for vinyl cutting.
GIMP is an open-source image processing program, with capabilities similar to Photoshop. Website here.
Below are the steps for converting a photograph into a black-and-white png ready to vinyl-cut.
Now this image can be exported as a png bitmap, imported into Inkscape, or other software, and traced to create a vector file for laser cutting or vinyl-cutting.