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Potential difference between two points is the work per unit charge to move between them: V_AB = -(∫_BA E·dl). Voltage is the practical term for potential difference. It is path-independent and depends only on endpoints, a consequence of the conservative nature of electrostatic fields.
Calculate potential difference for simple geometries by direct integration and by using potential functions. Measure voltages in circuits to build intuition for typical voltage scales.
You already know that the electric potential V at a point is the potential energy per unit charge placed there, and that the electric field E at a point gives the force per unit charge. Potential difference is the physically measurable quantity connecting these two ideas: it is the work done by the electric field per unit positive charge as that charge moves between two specific points in space.
The formula V_AB = −∫(B to A) E · dl captures this. Imagine carrying a small positive test charge from point B to point A along any path you choose. The electric field either helps or hinders your journey at each step; the total work done by the field per unit charge, accumulated over the entire path, is the potential difference V_A − V_B. Because electrostatic fields are conservative — a consequence of their Coulomb origin that you studied when learning about electric potential — this work depends only on the starting and ending points, not on the route taken. A straight path, a curved detour, and a zigzag all give the same answer. This path-independence is what makes voltage a well-defined property of a pair of locations.
The word voltage is the practical term for potential difference. A 9-volt battery maintains a 9 V difference between its terminals, doing 9 joules of work per coulomb of charge that moves from the negative to the positive terminal inside it. That energy is then available to drive current through an external circuit — flowing from the high-potential terminal through resistors, LEDs, or motors, losing energy along the way, and returning to the low-potential terminal. Ohm's law connects voltage and current: the potential difference across a resistor equals the current through it times the resistance, V = IR.
A subtle but important distinction: potential (V at a single point) is only defined relative to an arbitrary reference, typically chosen to be zero at infinity or at a ground node. Potential difference (ΔV between two points) is absolute and physically meaningful regardless of that reference choice — you cannot measure absolute potential with a voltmeter, but you can always measure the difference between two probes. This is why voltage is the natural language of circuits: every component has a voltage *across* it, and the sum of voltage drops around any closed loop must equal zero (Kirchhoff's voltage law). The path-independence you establish here is the foundation for equipotential surfaces, capacitance, and eventually the relationship between E and V via the gradient.