An eigenstate is one of the many possible states which may exist prior to quantum decoherence. According to quantum mechanics, a particle such as a photon, aimed at a mirror at a 45 degree angle, does not take the single path which is described by classical physics. In actuality, the particle takes every possible path off of the mirror. Each of those potential paths is an eigenstate. Once a conscious observer looks at this system, the particle "decides" which path it is going to take, and that eigenstate ends up being the one that is percieved.

The story of Schrodinger's Cat is one of the most well known anecdotes involving eigenstates. The cat in the box exists in eigenstates of both alive and dead until an outside factor (conscious observation, interaction with a classical system, etc.) causes the quantum waveform to collapse.