An inductor and a capacitor exchange energy back and forth — the capacitor storing it in an electric field, the inductor in a magnetic one. At one particular frequency their reactances are equal and opposite and cancel completely. That is resonance.
The square root has a practical consequence: to double the frequency you must reduce L or C by a factor of four. Tuning across a wide band therefore needs a large capacitance range, which is why variable capacitors in old radios were physically enormous.
Series and parallel behave oppositely
A series LC becomes a short circuit at resonance, limited only by the parasitic resistance — useful as a notch filter that swallows one frequency. A parallel LC, or tank, becomes a very high impedance at resonance, which is what makes it the frequency-selecting element in an oscillator or the load in an RF amplifier.
Q factor and bandwidth
Q measures how sharply the circuit selects. It is the ratio of the characteristic impedance √(L/C) to the loss resistance. High Q means a narrow bandwidth and a long ringing time; low Q means a broad response that settles quickly.
A tuned circuit with Q of 100 at 1 MHz passes a 10 kHz band. Real inductors rarely exceed Q of 200 because of winding resistance and core loss, and the loading of the stage that follows usually dominates anyway.