Plate count is a (theoretical) measure of the efficiency of your column. You know that in chromatography there's an equilibrium between the stationary phase and the mobile phase; during elution there's a transfer of molecules from the mobile phase to the stationary phase and back to the mobile phase, and so on back and forth down the length of your column--that's how the separation occurs.
The distance along the column that it takes to make one of these transfers between phases and re-equilibrate is called the "theoretical plate height." If it takes less distance, that means the plates are narrow and you have more of them in the column, which means more transfers (and therefore equilibria) between phases can take place, and your column is more efficient. That of course means better resolution and better peak separation for you.
If it takes a comparatively greater distance for one phase transfer event, the plates are wider and there are fewer of them, so your efficiency (and therefore your resolution) is not as good.
The relationship between theoretical plate height, plate count, and column length is
theoretical plate height x plate count = column length
plate count = column length/ theoretical plate height