Just a Wall!
Perhaps the simplest backdrop is the plain ol' boring wall. It's simple, it's clean, it works. If less is more, the plain wall is probably as minimalistic as you can go.
If you don't a super large wall, don't worry. Just pose your model or object closer to the wall. For head shots, you won't even need more than a 3 feet by 3 feet wall clearing.
Simple painted and wallpapered walls work too. My walls just happen to be white.
Fabric backdrops are simple, cheap, alternatives to professional muslin backdrops.
They're really convenient if you don't have a clean wall; just drape fabric over a door or hang it from the ceiling. And there's a lot of variety to choose from, so it's easy to change the color to whatever suits your photo.
Plus, most fabric backdrops don't cost you a thing: just recycle your bed sheets, a blanket, that couch throw, or even a large shirt.
The most common backdrop is the muslin backdrop. You can handpaint or tie-dye your own. Refer to a search engine to find tutorials on how to make your own muslin backdrops.
I purchased my muslin backdrop on Ebay for $35 as seen on the right.
Muslin backdrops are classic. If you are looking into portraiture, a nice handpainted muslin backdrop is a worthwhile investment.