Why does the current lead or lag voltage in an AC circuit with a capacitor or inductor?

In a pure resistor, current and voltage rise and fall together, so they are in phase. A capacitor and an inductor behave differently because of how they store energy. A capacitor opposes a change in voltage: it must charge up before a voltage appears across it, so the current flows first and the voltage builds up after. That is why current leads the voltage by 90 degrees in a pure capacitor. An inductor opposes a change in current because a changing current induces a back-emf that resists the change, so the voltage appears first and the current catches up later. That is why current lags the voltage by 90 degrees in a pure inductor. A handy memory aid is CIVIL: in C, I leads V; in L, V leads I.