TragicMonkey
Poisoned Waffles
Not a lot. It's just Time magazine's recognition of the person they deem most influential in that year. They don't have to be a good person, just influential. In 1938, the Person of the Year was Adolf Hitler.
So it's more or less completely irrelevant. Time is a large and influential periodical - or at least it has been for most of its history - so in terms of celebrity status it's pretty influential. But it doesn't really mean anything other than that.
The best pick was the year they chose the Inanimate Carbon Rod.