the cutoff frequency
Every filter needs a place to say here is where I start cutting. That dividing line between what passes and what gets blocked is the cutoff frequency, also called the corner or break frequency. It is the single most important number on a filter's spec sheet, the frequency that sets where the action happens.
By universal convention the cutoff is the -3 dB point: the frequency at which the output has dropped to 1/sqrt(2), about 70.7 percent, of the passband level. That fraction is not arbitrary; it is where the output power is exactly half, since power goes as voltage squared and 0.707 squared is 0.5. For a simple RC filter the cutoff is f_c = 1/(2 times pi times R times C). With R = 10 kilohm and C = 16 nF, f_c is about 1 kHz. On a Bode plot the cutoff is the corner where the flat passband bends into the sloping roll-off, which is why it is also called the corner frequency.
You choose the cutoff to separate signal from junk: set a low-pass cutoff above your highest wanted frequency but below the noise, or set an anti-alias cutoff below half the sampling rate. The honest caveat is that the cutoff is a soft transition, not a wall: at f_c the signal is already 3 dB down, signals just past it still leak through, and the filter also introduces its biggest phase shift right around the cutoff.
You want a low-pass with a 1 kHz cutoff. Pick a convenient capacitor, C = 10 nF, then solve R = 1/(2 times pi times f_c times C) = 1/(2 times pi times 1000 times 10e-9), which is about 15.9 kilohm. Use the nearest standard value, 16 kilohm, and the cutoff lands within a percent.
Pick a capacitor, solve for the resistor: the cutoff formula sizes an RC filter in one line.
Cutoff means -3 dB (about 70.7 percent), not zero output. The signal is already attenuated there, frequencies just past it still pass, and phase shift is largest right around the cutoff.