Most artists will use Photoshop, personally I prefer PSP9.0 as it seems a bit more technical and exact to me. Anyway, to actually get the graphics done, in whatever art package you use, it's a case of either having the skills, or having the software.
If you wanted say, a nice mountain to use as a backdrop to a game, then the easiest way to do that is to use a terrain renderer like Bryce. Often it's very quick to put a mountain together like that, with sky and all that stuff that you might need - compared to drawing from scratch, it's a different sport altogether.
It takes practice, and lots of experiments to see what works, and fall into your own groove, and establish a style that you can stick to. It's nice to stylise, especially with more tongue in cheek games, platformers etc are much more forgiving. See I think it's better to avoid settling for a sub-standard piece of work - if it doesn't look right, do something else. It's graphical polish that determines how professional the game looks, and that covers things like animation - you have to be strict with yourself on top of everything, honest and strict when it comes to artwork - one little ugly or miss-matched thing can ruin a whole scene.