
An eigenstate is one of the many possible states which may exist prior to quantum decoherence
Ok, I don't know what quantum decoherence means, but here's a better analogy.
The story of Schrodinger's Cat is one of the most well known story involving eigenstates. The story goes like this:
There's a cat inside a box, with a poison ready to be release anytime. Since you don't know when the poison is released, the cat is either alive or dead. Alive and dead is the eigenstates.
So what does this philosophical point of view means? It means that you can't predict where exactly a hurricane will touch land. You can only calculate the possible areas where it is going :). Until you open the box, you don't know if the cat is dead. Until the damage of the hurricane is done, you don't know what area will be the most damaged:)
Sadly, this is how tiny things operate--like the electron of an atom. You may never pinpoint the exact place where the electron is, but you can calculate the probability where the location is.
I saw that shirt by the way in one of my students. Pretty geeky.