I understand all the fuss about normal distribution.
a lot of real life events and their probability of occurence
follow a normal probability distribution.

But why the need for
a) student's t
b) chi square
c) f distribution

For example why do we need an experiment so we could obtain a
sampling distribution for the chi-square statistic?

or for example I read this on stattrek

But sample sizes are sometimes small, and often we do not know the
standard deviation of the population. When either of these problems
occur, statisticians rely on the distribution of the t statistic
(also known as the t score), whose values are given by:

t = [ x - μ ] / [ s / sqrt( n ) ]

where μ is the population mean?

HOW IS IT POSSIBLE THAT WE KNOW THE POPULATION MEAN BUT NOT THE
POPULATION STANDARD DEVIATION? This is what I don't understand.
If we know the population mean, SHOULDN'T we know the population
standard deviation?

Or sometimes it is the other way around, we know the population standard deviation but not the population mean...

you ask how can it be that we know the population mean but not the population standard deviation, well, usually we DON'T !

we make inference on the population mean, because we do not know it. and in order to do so, we need to know the standard deviation, but usually we do not know that too, so we use the sample standard deviation instead, and that's how we get the t-distribution.

as for the chi square distribution, we need it to make inference on the standard deviation and variance. The chi square can not get negative values, and neither can the variance or the standard deviation.

So, basically you are saying that we don't know population mean, μ. We speculate what population mean, μ is. And the reason we are guessing/speculating the μ is so that we could get t- statistics and through that we could use confidence interval to test to a certain confidence level that the μ that we speculated is actually the true population μ.

The same process for chi square but this time, it is to test population variance or population standard deviation?

Please confirm if I have this right...

Also, wonder if anyone has a real life example to help me grasp the concept better. or a example on the internet somewhere.

μ (mu) is the true population mean.
It is not possible to ever know the true population mean.

The common example is height. For a given county, there is at all times a true mean of the height of every person in that county.
No matter how many people we measure, or how often, we will never know the true mean for several reasons:

1) most importantly, one of three main sources of variability is measuring variability. No matter how accurate our tools or how fastidious our technique, our measurement will be wrong to some degree.
2) another of the main sources of variability is sampling variability. No matter how many people we measure, we are only measuring a sample, and the sample is not going to be an accurate representation of every member of the population.
3) the true mean can change any number of times at any given time.

We can establish a sample mean, and a sample standard deviation, and it will be wrong no matter what we do.
What we have to do is attempt to ensure that it is 'close enough' for the purpose at hand.

What counts as close enough depends a variety of things, and can be tested for in various ways.