In typing a term paper, what is the proper spacing after a period? I think I've read that one space is now acceptable.

The old typewriting rule of putting two spaces after a sentence-ending punctuation mark, like a period, came about because, on a typewriter, a space is actually the same width as every other character on the typewriter. Adding the extra space helps the reader determine that a new sentence is starting — and it's a little easier on the eyes, as well!
Nowadays, word-processing programs like Microsoft Word automatically insert extra space after a sentence-ending punctuation mark — not as much as an extra space on a typewriter, but more than the normal space between words. So if you're using a word-processing program, you don't need to add the extra space. However, some teachers are still sticklers for those two spaces, so you'll want to ask your teacher what his or her preference is — and do what your teacher says.