What is a one-bit ADC good for?

To give a basic example of how a 1-bit ADC can be used to obtain useful information from a waveform, take a look at this circuit. It uses a triangle wave to turn the information into a pulse width modulated output. This is a similar but simplified version of how other 1-bit ADC techniques work, by using a (usually fedback) reference signal to compare the input to.

Circuit

Simulation

Magnified Timescale View:

We can see from the top input waveform, the triangle wave is used to compare the waveform at different points through it's period. As long as the triangle wave is of a considerably higher frequency than the input (the higher the frequency the more accurate), this causes the comparator to output an average of high/low depending on the voltage level of the waveform.
To see how we can reproduce the original waveform from the PWM data, the comparator output is fed into a low pass filter, and out pops the sine wave again.

For further reading:

Delta-Sigma Converters
Successive Approximation ADC
Single Bit ADCs
Ramp Compare ADC (Counter ADC)

A one-bit analog to digital converter (A/D) is just a comparator with the threshold in the middle of the range. Usually you don't call it a 1-bit A/D though, although it is legitimate to think of it that way.

There are ways to make use of a comparator to ultimately get a higher resolution digital value. A delta-sigma A/D is one example. This keeps integrating the comparator output and comparing it back to the analog input. Over a number of bit times, the analog value is represented by the number of 1 bits out of the whole. Resolution is a tradeoff with time. Nowadays the bit rate can be in the multiple MHz range. For example, at 10 MHz bit rate, getting a 20 bit result (about 1 M counts) would take 1/10 second.

Another example is a "tracking" A/D. This contains a D/A and the comparator compares the D/A result with the analog input. If the comparator result is low, the D/A value is incremented, otherwise it is decremented.

Another name for a one bit ADC is a comparator. I can imagine 1 bit ADC may be enough for an application which needs to turn on/off a valve, switch, alarm if the signal goes above/below a threshold.