How do we actually obtain samples to form discrete time samples?
Assume you are speaking, those vibrations (sound) are converted into electrical signals(continuous time signal) by Mike. After that, a Sample &amp; hold circuit is placed, which stores the converted electrical signal inside a capacitor in the form of charge (now u get discrete time signal). 

Every signal is  sampled at some frequency called sampling frequency. Let the sampling frequency in this case be 8KHz. It means, at time t=0, you sample (read) the electrical signal amplitude (by using sample &amp; hold circuit) and hold it(store it) in a capacitor. You wait till some time and when time t=0.00125 seconds(1/8KHz, remember our sampling frequency is 8KHz), you again sample the electrical signal and store it . You repeat this process every 0.00125 seconds. Now you have got discrete version of your continuous time electrical signal (in fact speech signal).

In the time between every samples(0.00125 seconds), the charge stored in the capacitor is converted to digital binary values using special circuits called ADCs. These converted digital values and discrete signals in binary form(1s and 0s which may be stored or for further processing in a computer).

If you are interested, read about how sample and hold circuit works. You need some electronics background for that.