The process is not hard, but it can be time consuming. The short version is, basically, to make all the background pixels a single colour leaving only the sprite, which you can easily grab.
PaintShop Pro makes this very easy, with a colour-replacement tool (icon at left). Apparently a similar tool exists in Photoshop, called the 'magic eraser'. According to Twyst, 'it has the option to turn off contiguous pixels, so it'll replace all pixels of the same colour.'
Select one background colour, and a new colour that's not used anywhere in the game, and double-click the image to replace the colours. After doing this for all the background colours, you'll have nought left but the sprite.
First, isolate the sprite you want. It's easiest to work with a small area, and if you have the luxury of making the screenshot yourself, try and put the sprite on a background with few colours. Here I've simply dragged a selection around the ThunderForce ship:
You can see here the results after double-clicking once, turning all of one background colour to a bright, easy to see pink. You'll often get lucky, and have a game that doesn't use quite the same colour for a background black as a sprite black, and you can whip through this process without trouble. Occasionally though the sprite will have the same colour as the background, and double-clicking will make pink some pixels you would rather stayed un-pink. You'll have to watch for this, but it's easiest if you simply put the sprite on a neutral background. In this case (ThunderForce AC) I had to play to level 3 to find a background that worked.
It doesn't take long until you've got a pink box with a sprite in it.
Using PSP's magic-wand tool (I'm sure there's a photoshop equivalent but I don't know what it is) you simply select the pink area, then invert the selection, and presto: you can copy-and paste that sprite!
And here it is, the ThunderForce AC ship, which you can save and enjoy!
The Painful Bit
When ripping some sprites you can end up with a huge mess of pink bits that must be selected individually, as they're not adjacent to each other. This Felicia image was a massive, huge pain in the ass, 'cause every diagonal segment and curl of hair left me with an orphaned pink pixel. It took forever to select them all (there's no 'select all of this colour' option in PSP, dammit).