Your question is about real shape of the wave forms of the sampled signals. Assume that you have a waveform x(t) and you want to sample it. The first thing to consider is the sampling rate or the sampling frequency fs. This frequency is limited by the Nyquist theorem where fs must be greater or equal to double the highest frequency contained in the analog waveform.
The inverse of the sampling frequency is the sampling time Ts which is the time between two consequent samples. That is Ts=1/fs.
The samples must be ideally in form of delta pulses having pulse width tending to zero and an amplitude equals to that of the waveform at the sampling instant ; that is S(n)= x(nTs), where S(n0 is the sample at the time instant nTs.
Practically one has to take the sample in the shortest possible time and then store its value for the rest of the sampling time Ts. The time required to take the sample is called the aperture or acquisition time. The time interval in which one keeps the amplitude of the sample constant after acquisition is called hold time. The hold time is required by the analog to digital conveter as it needs time to convert the sample from the analog into binary digital form where input sample value must be kept constant.
So, practically the sampling time is divided into a short acquisition plus a much longer hold time.
To take samples and hold them one uses sampling gates and storage capacitor.
So, one has to on the gate for a sufficient period to charge the capacitor with the required sample value. This process takes a time called the settling time which is the time from the application of the gate on pulse to the time where the voltage on the capacitor reaches the steady state value with certain allowed error or margin.
So, the settling time must be as small as possible and defines the acquisition time.