EMF: chemical to electrical
PD: electrical to other
The electromotive force (not actually a force) is how much energy is converted from chemical energy in the battery to electrical in the circuit.
The potential difference is how much of that electrical energy is converted then to other forms of energy, like light from a lamp in a circuit.
By Kirchoff's second law (the voltage law), the sum of the p.d's around a circuit is equal to the sum of the electromotive forces.
Emf is the maximum energy that is produced be a source; terminal voltage is the usable energy you draw from that source. They differ due to internal resistance in the source.
EMF is an non-electrical, non-force (so all-in-all, it's a poor name!) effect that indicates the maximum amount of work that the source can do on a per coulomb basis. It is measured in joules per coulomb, which is equivalent to the unit "volt" but it is not a voltage in that it is not associated with an electric field.
For instance, a battery with an emf of 1.5 volts can produce up to 1.5 joules of energy per coulomb of emitted charge. This energy comes from the chemical reaction occurring in the battery.
A generator produces an emf through electromagnetic induction by means of changing magnetic flux. The emf of a solar cell comes from the radiant energy incident on it.
The reason the potential difference at the electrodes is not the same as the emf is the fact that the source will have internal resistance, which converts some of the energy into heat as the charges are moving toward the terminals. (Hence a battery may feel warm when current is drawn.)
In terms of a formula, if
So, in a case where the source is not delivering any current (open circuit), the terminal voltage and the emf are identical. This is what happens when you test a battery with an ordinary voltmeter. To measure the difference between terminal voltage and emf, you must test the source under load (drawing a current). This is what a load tester for your car's battery will do.
If the terminal voltage and emf prove to be quite different, it means the internal resistance of the source is quite high. If this is a car battery, it's time for a new one!