First up the is one i'm sure most people are aware of and have messed about with at some point 'Artrage' 2.5 is the free way to play with it the newer and bigger versions cost but as a simple tool for painting quick basic backgrounds it does the job, next to painter and photoshop it is very lacking but what we're looking at here is starting points.
'Alchemy' an amazing and very interesting tool, it isn't for finished art it's for concepts, it's open source with artists encouraged to add to it, they call it "the art program by artists for artists". The tools are odd and take some getting use to but it is great, it's very fluid and can be used as a doodling pad on the desktop were experimenting will lead you into area's you wouldn't have expected, 'Scribbler' is similiar but a bit one note in comparison it's for doodling and gives a cool effect and is very much an entry into digital art but 'Alchemy' can be downlaoded installed and added to 'Scribbler' is just an online sketching tool.
'Pencil' is an open source program that is great for classically animating digitally here's a little video of it in action, it's a great little free program that after a short getting to grips with period can give some great sketched looking animations or clean depending how much time you spend with it.
Going more advanced you can experiment with 3d, 'Blender' once again an open source program with a huge community online and loads of tutorials it's very user friendly and the interface isn't as rubik's cude as 'Maya' or '3d max'. Here and here are examples of it the first being a basic tutorial the latter an example of what is possible.
For those with an interest in modelling and not wanting to download an illegal version of zbrush we have 'Sculptris' it's good for modelling and experimenting with building 3d objects or characters/creatures here's a demo video, a great place to start with 3d modelling as it offers a very simple interface and a more hands on feel to the creation not moving dots on screen like in other 3d programs it's like working with clay.
'Monkey Jam', is used to capture drawings into a sequence with can be exported as a video file or image sequence, it offers layers so you only have to draw a static background/objects once and it'll composite them with the animated action, the newer version includes audio exporting and more. It's very simple and with a web cam/digital camera, some paper and and a will to draw you can create some cool animations, it also has potential to be used for stop motion if you've got a tripod handy for your camera.
A program that is build into Windows, 'Windows Movie Maker' can be used for editing it has filters and fades to add to your work as well as camera movement simulation. only draw back is it'll compress things down alot unless you explore to export options.
Finally if you wanna record sound or edit music 'Audacity' will be very helpful it has a decent amount of options for a free audio editor and is very easy to use, a little plug in (also free) will allow exporting of mp3 as well as wav files.