Inkscape is the vector editing program for Linux, competing head-to-head with Adobe Illustrator, Corel Draw, Xara X and is fully compliant with the SVG (Scalable Vector Graphics) format. Inkscape features affine transformations, Spiro Curves (Cornu spirals), Z-order operations, layers, pattern fills, node editing and alpha transparency just to name a few. Inkscape can also be extended with plugins.
Artists working with 2-dimensional images have probably heard GIMP (GNU Image Manipulation Program) which is a competitor of Photoshop. The features are almost exactly the same with a different interface. If you will be converting from Photoshop, then you may want to check out Gimpshop, which is a fork of GIMP with a Photoshop interface.
3D artists will be pleased to see Blender, which is the equivalent of 3DS Max. Blender can be used for creating cinema-quality movies to interactive animations, including modeling, rendering, texturing, skinning, non-linear editing, post production and even more. Artists who want to work with ray tracers will find Pov-Ray Modeler extremely handy.