Serifs are hard to describe, so I'll let Wikipedia do it:
http://en.wikipedia.org/wiki/Serif
Basically, they're the little bits hanging off the end of letters on a font. In print, it's believed that serifs allow the eye to more easily follow the flow of the letter forms, which makes the text easier to read.
Times New Roman is a serif font
Arial is a sans-serif font
Look at the lower-case "i" in both lines. In a sans-serif font, it's just a simple straight line with a dot above it. In the serif font, it has pedestal on the bottom and a little hook on the top of the straight line, those extra flourishes are the serifs. They don't make the letter look more like an "i", but they can make it easier to read.