Why does a capacitor connected to AC and high impedance act as wire?

 I answered this SE EE question yesterday.

Your previous question and this question are about the capacitor's basic property of maintaining a constant voltage across itself during rapid changes in current. Depending on how the capacitor is connected, we distinguish two main applications of this property:

1. Decoupling capacitor. In this application, one terminal of the capacitor is grounded and the input voltage source is connected in parallel, through a resistor, to the capacitor. Then, since its voltage is constant, the voltage at its other terminal is also constant. The RC integrating circuit from your previous question exploits this idea.

2. Coupling capacitor. In this application, the capacitor is connected in series (with one of its terminals) to the input voltage source. Since its voltage is constant, the (output) voltage at its other terminal follows the input voltage variations. AC amplifiers with capacitive coupling exploit this idea. There, coupling capacitors are charged to the bias voltage; so the input voltage is "shifted".