I have some thick boards that I'd like to make thinner. Generally I would thickness plane the board, but in this case I'd like to reduce the thickness of the board by about half (e.g. from 2" thick to 1" thick or 1" thick to .5" thick). Thickness planning would be a real waste of material in this case. Is there a good way to cut a board parallel to its face such that I end up with one board of a desired thickness and another board of the original thickness - saw blade width - desired thickness? What kind of surface finish can I expect on the new face of the two resulting boards? The boards in question are roughly 8" wide. I was thinking I could use a band saw with a fence, but this seems pretty sketchy since the blade isn't rigid. Can you slowly raise the blade of a table saw and use a high fence to accomplish this (doesn't seem very safe)?
The operation you want to do is called resawing, and is best done on a bandsaw.
Ideally, your bandsaw should be equipped with a 1/2" or wider, 2-4 TPI blade and a resaw guide or high fence that can be easily adjusted for blade drift (just in case you haven't yet mastered Alex Snodgrass' drift-free bandsaw setup technique).
The bandsaw will leave a pretty rough surface, so after you've sliced up the board into thinner boards, you'll need to run each one through the thickness planer or drum sander (or hand planes) to clean up the bandsaw marks. If you cut the original board into more than 2 boards, keep in mind that you'll need to flatten one face on each of the inner boards before thicknessing, so be sure to cut the inner boards a little thicker than the outer boards for this reason.
If your boards are sufficiently thick, you can also resaw on a table saw, but it's much easier to resaw unsafely on a table saw and you end up wasting a lot more wood because of the much thicker kerf (vs. a bandsaw). If you do this, you should use a rip blade rather than a combination blade or crosscut blade; otherwise you'll have to take much shallower passes to avoid burning. Generally you shouldn't resaw all the way through on a table saw--in fact, most 10" table saws can only resaw about 6-7" wide boards on their own anyway. One resawing technique is take several passes cutting progressively deeper kerf in each long edge with the table saw, then finish resawing the middle on the bandsaw or with a hand saw.
You can also skip power tools entirely and resaw with a hand saw.
It's called resawing, and generally it is done with a bandsaw. There are blades that can make this process easier. Getting the largest (wide) blade you saw can handle is the best way, at least for wide boards.
I've used fairly small boards and cut them in half, 3/4" in half with thinner blades. but you get a lot of wobble, especially if you don't have a fence. This worked fine for me since I only needed 1/4" from each half and thickness planed them down.
But with a fence and a resawing blade you should be able to do a decent job, the biggest trick is to not force the board through, that is really what causes the blade to wander. Of course once you have it set up, it is a good idea to run a test piece through to find any issues your setup or technique could cause.
Using a table saw, while possible I would highly recommend NOT doing it.