Just like with a mass and a spring, energy oscillates between the electric field of a capacitor and magnetic field of an inductor. The natural frequency of the electromagnetic oscillation is simply
ω = 1/√LC

In your activity, you gave the LC circuit a pulse and watched the oscillations on your oscilloscope. Because there is always some resistance in any circuit, you saw that the oscillations grew smaller with time, but the frequency of those oscillations was always the same, even if you changed the input frequency.

However, suppose you have a circuit where you are constantly driving the LC components with an alternating voltage, forcing them to oscillate at a certain frequency. They still have their own "natural frequency" but that is overridden by the "driving frequency".

What happens when that driving frequency matches the natural frequency? Because the LC components want to oscillate at their own frequency and are now being driven at that same frequency, they will produce oscillations of a very large amplitude. That's called "Resonance", and exists for electronic circuits as well as mechanical systems. And that condition is simply
ω_d = 1/√LC where ω_d is the driving frequency.

If we have an LRC series circuit (the R is from the light bulb), we can drive the circuit with a function generator capable of a wide range of frequencies.

The function generator puts out the same voltage amplitude at all times, but it's not enough to light the bulb. But if you put in a frequency that matches the natural frequency, the voltage in the circuit is "amplified" by the resonance effect and the light bulb shines brightly. If you change the inductance by removing the iron core, you go "off resonance" and the bulb dims. If you add capacitance, you again move the natural frequency away from the driving frequency and the bulb dims. All this happens without changing the input voltage.

This is how a radio works. The old radio I brought into class is basically an LCR circuit where you change the capacitance to match the natural frequency of the circuit with the radio station frequency you want to listen to, which is the driving frequency.