Capacitors and inductors are important energy storage devices in electronics. They store energy in different ways, and they are often used in signal filtering.
The capacitor is easily modelled as two plates parallel to each other with a gap containing a dielectric or vacuum in between. Because of this setup, a charges can be stored on the metal plates, and a voltage can be maintained. This potential, when multiplied by the charge, gives the amount of energy stored in the capacitor.
The capacitor's ability to filter signals results from its ability to pass high frequency and block low frequency. At a low frequency, based on the model above, the capacitor looks like an open circuit or disconnected wire. At a high frequency, because of the changing electric potentials, the capacity actually looks like a short circuit.
The inductor is often looked at as the opposite of a capacitor in terms of filtering applications. Due to Faraday's and Lenz's laws, the inductor, as effectively a loop of wire, tends to oppose changes in current. So, when a signal has a low frequency (is not changing direction often), the inductor looks just like another wire. However, when the signal has a high frequency, the inductor actually blocks the signal!
Inductors do share a property with capacitors, though, in that they also store energy. However, the energy stored is magnetic, as opposed to electrical. As a result of electromagnetic induction and the properties of the materials around which the inductor is wrapped, the inductor can maintain a magnetic field and the energy therein.
These energy-storing and filtering properties are used often in electronics. Two important applications here are oscillators, where electric and magnetic energy is passed back and forth between a capacitor and inductor for certain frequency signals, and filters, where "noise" is kept out of a signal to ensure the proper signal is received.
I hope that helps!