How do we actually obtain samples to form discrete time samples? I'm in my second week of an online DSP class and I'm still kinda lost on some of the concepts of what we are actually doing.  I'm hoping you guys can help clear up my understanding.  Please forgive me if what I'm asking makes no sense.

I'm reading about continuous time sampling, which I don't think I fully understand yet, and discrete time sampling.  But I guess to understand those concepts I really need to understand sampling.

I think sampling is gathering discrete data points from a continuous time signal?  But how?  How is this physically done?  What are we using to get the actual data to convert it into a discrete time sample?  Is there a good example I can look at to understand this more?

Right now the text book just shows math formulas, pictures of graphs of sines with various frequencies and amplitudes, and with dots representing discrete time samples vs continuous time samples - but no application of what is actually happening?  For example, is an instrument making a consistent sound, that is being captured through some microphone and producing the signal, which is then passed through some system that calculates data points?

Is a continuous time signal just a signal being actively captured and a discrete signal just data points over a certain period of time?  How and when do I apply the mathematics we are learning about (as we haven't needed to really do so yet)?
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.