The IAU has a three part definition - which controversially excludes Pluto.
The three criteria are:
1. Is in orbit around the Sun, not another body.
2. Has a hydrostatic equilibrium shape, normally (but not always) round.
3. Has cleared its orbit of other objects and debris.
So, what's the problem with this?
We can't apply it to exoplanets. First, we'd have to change the first line from "Sun" to "a star" - which is easy enough. However, we can't tell what shape an exoplanet is, yet, nor can we tell if its cleared its orbit.
A man named Jean-Luc Margot, who's a professor at UCLA, has proposed a solution.
He's done the math to allow us to make an educated guess as to whether two and three apply based off of:
1. The planet's mass.
2. It's orbital period.
3. The age of the system it's in.
As far as we can tell it works, but we are working off a sample of eight, so the accuracy might be questionable. The theory, however, seems sound.