Serifs are referred to as such because of the small details that extend from the ending shapes of the characters; the origin of the word itself is obscure, various explanations have been given but none has been accepted as resolute.
Their origin can be traced back to the Latin alphabets of Roman times, probably because of the flares of the brush marks in corners, which were later chiseled in stone by the carvers.
They generally give better readability in print than on a screen, probably because of the better definition and evolution of the former in hundreds of years, while the latter technology is, on an evolutionary path, a newborn.
With the latest technologies and the high definition monitors that can rival the print definition, multiple scientific studies have been found inconclusive, showing that there is no discernible difference in readability between sans and serifs on the screen and as of today they are both used on the Web.
Within this general definition, there are multiples sub-families, as Old Style or Humanist.
Bodoni and Didot are certainly the most famous typefaces in this family.