The big split in the software world is between Vector and Raster type programs.
Vector programs draw everything by storing the equations for the lines you draw mathematically in memory. This is cool because you can get the same clarity at any level of resolution (crudely, "zoom')
Raster programs take each pixel of the image and assign it a color value. They are great for drawing images with detailed textures, or photographs.
There are excellent, free software packages in both flavors. In the raster world, the leaders of the pack are Adobe Photoshop ($$$) and GIMP (Free). In the vector world, there are Adobe Illustrator ($$$) and Inscape (free). It's possible to produce great maps in any of these programs, and each has its strengths and weaknesses. There are also many other programs that people can use. In the end, you will probably end up with at least one good vector program, at least one raster program, and several specialized "utility" programs for certain things.
I mostly use Adobe Fireworks. . . it is mostly a vector program but does an excellent job at applying raster effects to what's drawn. But this is a pretty uncommon choice. Xara Extreme is cheaper and is possibly more powerful for the same type of program.
Then there are programs like Campaign Cartographer - dedicated for the fantasy enthusiast cartographer. I am not an expert in it and will suggest looking at tutorials (for all programs, really. . . )
YES, it is very possible to draw usable dungeon-encounter maps on a computer! Not being a gamer, though, I wouldn;t know where to begin for that.