I think Kevin's question is expressed in terms of time, and should thus be replied to in terms of time. I reckon the trouble understanding reactive components (like capacitors) for most people lies in the difference between understanding their behaviour in terms of frequency and understanding them in terms of time. The two approaches are so far removed from each other that you might think we are talking about different components altogether.
In the time domain, a capacitor's voltage changes at a rate proportional to the current flowing through it. Thus:
Firstly, the capacitor's voltage cannot change instantly. It can only gradually approach some value, as it charges or discharges. The higher the time constant (R x C) of an RC circuit, the slower the rate of voltage change across the capacitor. If an input signal changes slowly enough, the capacitor in an RC circuit is able to charge and discharge quickly enough to keep up with input changes. If the input changes too quickly though, the capacitor cannot charge or discharge fast enough to follow the input. This inability of the capacitor's voltage to swing quickly enough results in its voltage being an attenuation of the input.
Secondly, intuitively, it can be seen that if the input changes significantly faster than the capacitor can follow, fluctuations of voltage across the capacitor will be negligible compared to fluctuations in input voltage. Thus it is the resistor that is dominant in determining the current through the network, and so the current is roughly proportional to the input voltage. This means that the rate of change of the capacitor voltage is proportional to the instantaneous input voltage. This is the cause of waveform distortion (not harmonic distorion).
So, in a low pass RC circuit, the output is the capacitor's voltage, whose rate of change is (nearly) proportional to the instantaneous input voltage, and the circuit is said to integrate. Conversely, with the resistor and capacitor swapped to form a high-pass filter, the ouput is the resistor's voltage, and the circuit differentiates, so that the ouput at any instant is (nearly) proportional to the rate of change of input. Read that again.
The upshot of all this is that not only does the RC network attenuate, but it also distorts, by either integrating or differentiating. So, for a low-pass RC circuit, a square wave input (whose period is well below the R x C time constant of the circuit) will appear heavily attenuated at the output. It will also be distorted into a triangle wave, because the alternating high and low input voltages are causing the capacitor to charge and discharge (nearly) linearly - otherwise known as integration.
For that same circuit, fed with a nice sinusoid of period significantly lower than R x C, the sinusoid is integrated. The integration of sin(x) is another sinusoid, phase shifted by -90 degrees (in other words, an upside down cosine, or -cos(x)). A high-pass RC filter would phase shift by +90 degrees, to yield a cosine. Try plotting a sinusoid, and then the rate of change of that sinusoid, and you'll see this effect clearly - two sinuoids out of phase with each other by 90 degrees.
An interesting experiment to demonstrate this is to connect an oscilloscope in X-Y mode to the circuit - channel 1 to the input, and channel 2 to the output. This yields a wonderful ellipse, or nearly a circle if you choose your input signal frequency and channels' vertical scales properly.
In summary, when feeding a simple RC network with a sinusoid, you get another phase-shifted sinusoid across the capacitor. With non-sinusoidal input waveforms, the capacitor voltage is always a distortion of the input. The amount of distortion depends upon how far the capacitor's rate of charge/discharge is exceeded by rate of change of input. Periodic signals whose periods are significantly smaller than the time constant R x C will appear in attenuated and integrated form across the capacitor, and in attenuated and differentiated form across the resistor.
Understanding an RC network in these terms (the time domain) permits an understanding of how timing circuits (like the 555 IC) work, but is not really appropriate for understanding filter applications. Those are best described in the frequency domain, because the input waveform is rarely some nice square, triangular or sinusoidal form.
