I recently checked my Erdös number and it seems that it is 4.
But what does that mean?

First, Paul Erdös was a mathematician and he published really a lot of scientific papers (>1500) together with a large and broad number of peers.
If you want to study connectivity among researchers, you could plot each author as a node in a graph and connect those authors that have published a paper together. In this context, the Erdös number is defined as the distance between Paul Erdös and a given researcher.

Thus, what does an Erdös number of 4 mean?
Erdös himself is given the Erdös number 0. Researchers that have published directly with Erdös get an Erdös number of 1. Researchers that have published a paper with direct collaborators of Erdös get an Erdös number of 2… Thus, a 4 means that I have published a paper together with someone (Erdös number=3) who published a paper together with someone (Erdös number=2) who published a paper together with someone (Erdös number=1) who published a paper with Paul Erdös.

Erdös numbers range up to 15, the median is 5, and the mean is 4.65.

By the way, there exists also a variation, the (Kevin) Bacon number. The idea is very similar: Actors who played together with Kevin Bacon get a Bacon number of 1…
Interestingely, there are also some people on the planet that have an Erdös–Bacon number, which is defined as the sum of the Erdös number and the Bacon number. Although these people are rare, there are some, e.g. Daniel Kleitman, a mathematics professor at MIT. His Erdös number is 1 and his Bacon number is 2, thus his Erdös–Bacon number is 3.