Cutoff Frequency and the First-Order Low-Pass
Lows pass, highs are blocked
The graph's horizontal axis is frequency on a logarithmic scale, and the vertical axis is the gain, the ratio of output size to input size. Raise the frequency from low to high: at first the output nearly matches the input, then it shrinks; find the cutoff frequency where the gain drops to 1/√2, that is, −3dB. The lower tick on the vertical axis marks exactly that height.
The capacitor makes a frequency-dependent divider
Connect a resistor and a capacitor in series and take the output across the capacitor, and the two form a voltage divider that shares the voltage. But the capacitor's impedance, 1/ωC, is inversely proportional to frequency. So at low frequencies the capacitor takes almost all the voltage and the output resembles the input, while at high frequencies the capacitor's share shrinks and the output withers.
The cutoff frequency ω_c = 1/RC
The boundary is the frequency at which the resistance and the size of the capacitor's impedance become equal. That point is ω_c = 1/RC, and because the two impedances are equal in size but at right angles, the output falls to 1/√2 ≈ 0.707 times the input. In terms of power that is half, so this is called the −3dB point, and in phase the output lags the input by exactly 45 degrees.
Above cutoff it rolls off at a fixed slope
Above the cutoff, every time the frequency rises tenfold the gain falls to 1/10, that is, by −20dB. This constant slope is the signature of a 1st-order filter. This plot draws the gain itself on the vertical axis, so it looks curved, but in a Bode plot with the vertical axis in dB as well, a flat straight line and a line falling at −20dB/decade meet in a corner, and that corner is exactly the cutoff frequency.
Back to the first screen
As you raised the frequency, the gain started out flat, bent near the cutoff, and fell; at exactly ω_c = 1/RC the output was 0.707 times the input, −3dB. Above that it fell steadily by −20dB per decade. The fact that the same circuit can pass or block depending on the frequency sent through it is summed up in a single cutoff frequency.