Yes, you will need a DS FLash cart like the M3 or R4 as Glyde Bane stated. There are other brands, but these two (which are pretty much the same hardware) are probably the best for you're money out there atm. They are highly compatable with pretty much all software out there and you can order one for about $30 if you look around the net.
What there are, are little DS Carts, that hold SD Mini memory cards.
Basically they come ready to go, though it is advised to download and update with the software from there page.
Anyhow, once you have your cart and your memory card, and you have the software on it up to date (which is as easy as copying files from one folder to another basically), you will then be able to copy DS software onto your memory card. Anything with a .NDS extention will run, just like an EXE in windows or what ever. It's really quite simple.
As for DSDOOM or DOOMDS, what ever they are calling it now. Well I'm not sure as to where the main project page is because there are at least 2 or 3 people out there messing with the sources, so I put up the best version on my page here for now:
http://files.frashii.com/~sp00nz/Doom/files/DS/DSDOOM/
With this you will merely extract the archive so that all the files are in a folder called DSDOOM or DOOM or what ever. You will then need to copy over your commercial DOOM iWAD files I.E. DOOM.WAD UDOOM.WAD, DOOM2.WAD, TNT.WAD, PLUTONIA.WAD. You don't need them all, just the ones you want to play.
Once you got all this in order you merely copy of the DOOM folder to your Memory card. Then take the memory and put it into your M3 or R4, boot up your DS, go into the Folder and launch DSDOOM.NDS.
EDIT: I uploaded an early attempt to get Quake DOOM running on the DS. I'm unsure if it will crash on you, or what bugs it may feature atm, but you can use it instead of a DOOM iWAD if you don't have one.
I got a little section about the DS here:
http://starwarslegacy.net/NDS.aspx
As for the Star Wars DS project (as it is called for the moment), you will not need your original DOOM files to play. So if you don't have DOOM, no worries.
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Oh and as for the screen shots. Those are not in game, they are merely old screen grabs of some Jedi Knight models I took about 2 years ago. So what you are looking at are merely old bases for my sprites, though in the GzDOOM project sprites retain there original quality very well as I don't have to shrink them down too much to get them to appear correctly in the engine, also I don't have to limit them to a 256 color pallette. The ones you see there are merely fillers, though I'll have some in game screens up soon enough, then you can see how well they actually appear in game.
Now as for the DS version, the gfx are a far cry from looking like that...
...imagine those images, now imagine them as viewed in the original Dark Forces, then you'll have an idea how it pans out on the DS.

He who fights with monsters might take care lest he thereby become a monster. And if you gaze for long into an abyss, the abyss gazes also into you.