The A and B channels have amplifier / attenuatators and connect to analogue to digital converters so they sample the waveform and represent the analogue values captured. So a sine wave looks like a sine-wave.
The C and D channels connect using digital logic levels direct into the sampler which then registers them as either high or low logic levels on each sample point, effectively like a logic analyser. The Quad will treat any level below say 1V as low, and values above 2V as high. One should avoid connecting the C and D inputs to anything other than logic levels like CMOS or TTL.
C and D can be used to capture the timing relationships of logic signals and can also be used as triggers in conjunction with the A and B channels.
Some people also use A and B as quasi digital inputs to get effectively a 4 channel logic analyser.
Analogue probes pass the signal either directly x1 or through a 10x attenuator in the probe. The x10 setting has higher impedance and lower capacitance (typically 10MOhm) to minimise the load on the circuit under test. Digital probes are typically just a simple direct connect as they need to pass the logic level straight through to the input. An analogue probe in x1 mode acts pretty much the same as a digital probe.