Looking back at the voltage divider, it is a circuit where the voltage across the second resistor is proportional to the ratio of the second resistor divided by the total resistance of the divider. Since ohms are used for the calculations, we can replace the second resistor with a capacitive reactance and still get the same results.
In a purely resistive voltage divider, when both resistances are equal results in an output voltage that is half. With a reactance, there's a frequency that will set the reactance to be equal to the resistance, thus only half of the alternating signal will be available for further processing.
There's another widely used frequency where the signal starts to be noticeably attenuated if seen from the output of the divider. This frequency corresponds to the point where the reactance will cause an output of about 70% (0.7071x approx, which is the square root of 1/2) of the original signal being delivered to the output; this frequency is called the cutoff frequency.
The cutoff frequency in a filter is used for both blocking higher or lower frequency signals. If blocking lower frequencies you can think of it as the point where the filter start to conduct a large portion of the input.
This basic circuit is the basis for most passive filters. One of the disadvantages of this simple design is that it only allows for attenuation of a signal, but other times it is of more use to amplify a given range of frequencies and block others, instead of just attenuating.