What does direction of EMF mean?

When you change the magnetic flux through a surface bounded by a closed loop an emf is induced in the loop. That means that if a test charge is taken once around the loop in one direction, the work done per unit charge on the test charge by the electric field that curls round the changing magnetic field is, by definition, the emf. The direction of the emf is that of the electric field. If the loop is an actual loop made of an electrically conducting material, then charge will flow round by itself, in the direction of the emf. In this case the work that has to be done against resistive forces is supplied by the emf.

No potential differences are involved. In fact the whole notion of potential difference is inapplicable to this situation. Potential differences arise from static or quasi-static charges, and you don't have these here.

However if you had a conducting loop with a gap in it, and certain configurations of changing magnetic flux, the emf can make electrons pile up on one cut end of the loop, leaving the other end depleted of electrons and positive. We would then have quasi static charges and a potential difference between the ends. [Electrons will stop flowing, even if there is still changing magnetic flux, when the electric field in the ring due to this changing flux is balanced by the opposing electric field due to the charge separation.] Note that even in this case, where there is a pd, the direction of the emf may be defined independently of the notion of pd.