A resistor and a capacitor form the simplest useful filter. Which output you tap decides its character: take the signal across the capacitor and you get a low-pass response, take it across the resistor and you get a high-pass one. Both share the same corner frequency.
At f_c the output amplitude has fallen to 1/√2 of the input, which is −3 dB, and exactly half the input power. This is why the cutoff is often called the half-power point rather than the point where the filter “stops” — it does not stop, it rolls off gradually at 20 dB per decade.
A first-order filter is gentle
One decade above the cutoff a low-pass filter attenuates by only a factor of ten. If you need a sharp transition — an anti-aliasing filter ahead of an ADC, for instance — cascade stages or move to an active Sallen-Key or Butterworth design. Cascading two RC stages naively does not simply double the slope, because the second stage loads the first; buffer between them with an op-amp.
Choosing R and C
Only the product sets the frequency, so 1 kΩ with 160 nF and 10 kΩ with 16 nF give the same corner. The split matters for impedance: a low R loads the source hard, a high R makes the filter noisy and sensitive to the input bias current of whatever follows it. For general signal work, keep R between roughly 1 kΩ and 100 kΩ.
Phase, which is easy to forget
The filter shifts phase as well as amplitude — 45° at the cutoff, approaching 90° far past it. Inside a feedback loop that shift eats phase margin and can turn a stable amplifier into an oscillator.