How long does a capacitor take to charge?

About 5 time constants, or 5 x R x C seconds. At that point it holds about 99.3% of the supply voltage, which counts as fully charged. With a 10 kΩ resistor and a 100 µF capacitor, the time constant is 1 second, so the capacitor is effectively full after 5 seconds.

The voltage follows V(t) = V x (1 - e^(-t/τ)), so it rises fast at first and slows as it gets close to the supply. To reach a target below full, solve for t: t = -τ x ln(1 - Vtarget / V).

If a constant current source charges the capacitor instead of a resistor, the voltage climbs in a straight line, and the time is t = C x ΔV / I. The capacitor charging time calculator works out any target voltage, and the RC time constant entry explains the 63% rule.