Current

Current is defined as the rate at which electrical charge flows through a given area of a circuit. The SI unit for current is amperes(also called amps), labeled A usually. 1 Ampere is defined as 1 C/s. If the rate of flow is constant, you can express current as just the change in charge flowing through over a change in time. If the flow isn't constant, it boils down into a derivative.

Current can flow in either one direction flows, known as direct current(DC), or in periodically reversing directions, known as alternating current(AC). The main types of circuits discussed on this site are DC circuits, where current only flows one way.

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All currents produce their own considerable magnetic fields due to the fact that current is effectively constituted of charges moving with high velocities.

Voltage is just another name for the difference in electrical potential, or potential difference, between two points in a circuit. This can also be expressed as the work per unit charge required to move a given charge between two points. Batteries apply voltage to the circuit, known as the battery's electromotive force(emf). EMFs can also be electromagnetically induced by a change in magnetic flux connected to the circuit. Another way to look at voltage is that it's effectively the driving force behind the charges, mainly electrons, that flow as a current through a circuit.