It happens - rarely, but it happens - to be the case sometimes when a man likes his given name enough to name his son after him, and the son likes the name just as much and decides to name his own son the same name1. In this case, suffixes are added to each man's name to differentiate him from others in the family tree: Joe Bloggs Sr. -> Joe Bloggs Jr. -> Joe Bloggs Jr. Jr. Now, hang on, that sounds mildly ridiculous. Wouldn't it be more intuitive to number them? Yes. So Joe Bloggs Jr. Jr. becomes Joe Bloggs III, pronounced "Joe Bloggs The Third". It doesn't stop there: Roman numerals can theoretically be used to stretch out a family tree indefinitely (though I'm yet to see a Joe Bloggs MMLXV).
A more common usage of this notation is in any given monarchy. Take the British monarchy (just for shits and giggles). They had Kings Henry III, Edward III, Richard III (who was, of course, written into fame by a sharp fellow), William III and George III. There hasn't yet been a female "third"; only queens called Mary and Elizabeth can be "thirds" until someone else (i.e. a "second") comes along named Jane or Anne. Of course, in monarchies this notation has been well and truly known to exceed III - we have, for example, the French monarch Louis XVIII (the "eighteenth", or rather "dix-huitième"). That said, monarchs with suffixes aren't necessarily straight after one another - Queen Elizabeth I reigned some 400 years before Queen Elizabeth II. Note also, if you're lucky, you might even see some celebrities with Roman numeral suffixes - one example that springs to mind is Frankie Muniz, a.k.a. Francisco Muniz IV, of Malcolm In The Middle fame.
1 Now hang on, waverider37, you cheap sexist bastard. What about a woman and her daughter? Well... it's possible, but a lot less probable. It may happen exceedingly rarely that a female may be a "third", but a very precise set of circumstances has to occur:
- Jane Bloggs I must give birth to Jane Bloggs II. Easy.
- Jane Bloggs II must not marry, or must keep her maiden name. Harder.
- Jane Bloggs II must give birth to Jane Bloggs III, thus giving her the exact same name (sans suffixes) as herself and her grandmother.
Generally, the middle step there is trickiest (but not impossible). Most women prefer to take their husbands' names rather than keep their maiden names - a reasonable assumption is around 95%, but I'm yet to back that up - so giving birth to Jane Bloggs III is impossible for Jane Foo (née Bloggs). Or, rather, extremely bad style. (In this case, Jane Bloggs III would technically be Jane Bloggs II if she, for example, chose to legally change her name.) Additionally, I know very few unmarried couples who use solely the woman's maiden name as the surname for any offspring. More often than not, they use both surnames hyphenated, which also means that the daughter isn't named exactly the same as her grandmother.
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