An electromotive force (emf) is the "force", measured in volts, that is produced by interaction between a current and a magnetic field, at least one of which is changing. Since the word "force" now has a very specific meaning in physics, and an emf is not a force in this sense, the expansion of the acronym is considered obsolete; or at best, an embarrassing historical artifact.
The emf describes the electrical effect of a changing magnetic field. In the presence of a magnetic field, the electric potential and hence the potential difference (commonly known as voltage) is undefined (see the former) – hence the need for distinct concepts of emf and potential difference. Given this emf and the resistance of the circuit, the current can be computed with Ohm's Law.
Emf is often used as a synonym for any potential difference, irrespective of the source of the potential difference. (E.g., a battery, charged capacitor, or electret might be the source.) This usage is considered obsolete.
See also: electrochemical potential