How are Serif and Sans-serif fonts different, and when should one use one over the other?
What is the difference between these fonts, and what are some typical examples of why one might be used over another?
Serif Vs Sans Serif
(a picture speaks a thousand words) Read @Calvin's answer for explanation.
Serifs are the usually perpendicular projections found on the termini/endpoints in type. For instance, a capital "I" is usually rendered with 2 crossbars. Those are serifs.
Sans-serif just means "without serif." The definition of serif / sans-serif typefaces should be self-explanatory.
Another name for serif is "roman"; likewise, sans-serif typefaces may also be referred to as grotesque / grotesk or gothic.
There are also different types of serif, such as slab serif—also referred to as Egyptian, mechanistic, or square serif—versus bracketed serifs.
Additionally, there are some typefaces with serifs that are still considered sans-serif. Bell Gothic is an example of this. And, lastly, some typefaces have what are called petit-serifs ("small serifs") or semi-serifs.
As Charles Stewart noted in his comment, "roman" is also used to refer to the upright straight-lined typestyle (as opposed to italic) reminiscent of classical Roman chiseled type—from which serifs are also derived. "Roman" (by itself) is commonly the base font of a typeface or font family, but there can also be a "bold-roman", "black-roman", as well as "roman-oblique", which is slanted at an angle but maintains the same glyph shapes as the base font.