I have been reading Signals and Systems (2nd ed.) recently. Chapter 7 is about sampling. Oppenheim uses discrete-time signals and continuous-time signals to explain sampling.
Sampling discrete-time signals, i.e., using only every $N^{th}$ sample of a sequence of samples, is useful for efficiently processing, transmitting, or storing information, if we can be sure that the sampling rate can be reduced without significant loss of information. This process is called decimation. Whether the sampling rate can be reduced or not depends on the frequency content of the signal.
Example: if the signal has a low pass characteristic with no significant components above $f_s/4$ (where $f_s$ is the sampling frequency), the sampling rate can be reduced by a factor of $2$, i.e., every other sample can be thrown away without losing information.
There is a difference between a quantized signal, i.e. taking only integer values, and a sampled signal, i.e. the value of which is only looked at on spaced instants.
A physical signal is usually continuous in values and in time.
An A/D converter turns a physical signal in a quantized one; if the converter is asynchronous, it generates a time-continuous signal.
Discrete-time/continuous-value signals are created by sample & hold circuits.
And digital signals processed by computers are usually both quantized and time-discrete.
unless it's a flash, the A/D needs time to figure out what the signal value is. what's it supposed to do while the continuous-time signal is changing as it is determining the signal value?
then with processing, how to do digital processing in a continuous-time context? totally parallel processing? lotsa gates. one of the promises of Digital Signal Processing over analog signal processing is that you need not change and rewire the hardware to change the algorithm. how're you gonna do that with a flash A/D and continuous-time processing?
