Voltage
Voltage is the electric potential difference between two points, the push per unit charge that drives current, measured in volts.
Definition
Voltage is the push that makes electric charge move through a wire, and the bigger the voltage the harder the push, much like water pressure: high pressure does not mean water is flowing, it means water would flow strongly if the tap were opened. It is the potential difference, the energy given to each unit of charge moving between two points, measured in volts, where one volt means one joule per coulomb; a battery or generator creates a voltage that pushes current through a circuit according to Ohm's law, $V = IR$, and voltage is always measured between two points, such as across a bulb or between a battery's terminals. Formally, voltage is $\Delta V = V_B - V_A = -\int_A^B \vec{E}\cdot d\vec{l}$, the work per unit charge done against the electric field, so $\vec{E} = -\nabla V$; an ideal source provides an electromotive force $\mathcal{E}$, while a real source delivers terminal voltage $V = \mathcal{E} - Ir$ because of internal resistance $r$, Kirchhoff's voltage law states that potential differences around any closed loop sum to zero, and power delivered to a circuit element is $P = IV$.
Example
A small AA battery gives $1.5$ volts, a car battery gives $12$ volts, and a wall outlet in the United States gives about $120$ volts, which is why outlets are dangerous and batteries are not; a $9\ \text{V}$ battery driving $0.5\ \text{A}$ through a small motor implies a resistance of $R = V/I = 9/0.5 = 18\ \Omega$, with every coulomb of charge delivering $9\ \text{J}$ to it. A battery with $\mathcal{E} = 12.0\ \text{V}$ and $r = 0.05\ \Omega$ supplying a starter motor drawing $150\ \text{A}$ sags to a terminal voltage of $V = 12.0 - 150 \times 0.05 = 4.5\ \text{V}$ during cranking, dissipating $I^2r = 1{,}125\ \text{W}$ internally, which is why dashboard lights dim when an engine turns over.
Key Insight
A bird sits safely on a $10{,}000\ \text{V}$ power line because both its feet are at the same voltage, and current needs a difference, of which there is none across the bird. High voltage is also what makes long-distance power transmission possible: for fixed power $P = IV$, raising $V$ to $500\ \text{kV}$ cuts $I$ and slashes $I^2R$ line losses, which is why transformers exist to trade voltage for current at each end.