Electromotive force

The name is a historical accident and the CED keeps it only as an alias for a potential difference. Emf has units of volts. A source of emf, a battery or a generator, maintains that difference between its terminals, and the charge already present in the circuit moves in response.

On the AP equation sheets emf is printed as a script E. The AP Physics 2 sheet gives it no equation at all in the Electricity group: the symbol appears over in Magnetism, in $|\mathcal{E}| = |\Delta \Phi_B / \Delta t|$ and $\mathcal{E} = B \ell v$. The AP Physics C: E&M sheet adds $\mathcal{E} = -L \, dI/dt$ for an inductor.

Emf against terminal voltage. They agree only when no current flows. A real battery has internal resistance, so once current is drawn, part of the emf is spent inside the battery and the potential difference you would measure across its terminals is smaller than the emf. AP Physics 2 defines a battery's emf as the potential difference measured across the terminals when there is no current in the battery, and treats batteries as ideal unless a question states otherwise.

In circuit bookkeeping an emf is a rise in potential when you traverse the source from its negative terminal to its positive one, which is how it enters Kirchhoff's loop rule.