what is the real Intuition behind the Standard deviation A typical statistics course will define the standard deviation as "the average of the difference between the data set and the mean ".
So if we tried to describe the definition mathematically we should derive this equation (∑(|x-mean|))/n
while, the known law for standard deviation is sqrt((∑(x-mean))^2⁄n)
which doesn't make sense to me why we take the square root of all the variance function[(∑(x-mean))^2⁄n] instead the square root of only the squared part.
Example if I have a list of 2 numbers [14,6]
the arithmetic mean = 10
the average of distance of each value from the mean should = 4 , (14-4 = 10, 6+4 = 10)
while the standard deviation law will calculate the value 4 x sqrt(2) ( 14 - 4sqrt(2) != 10)
So, regarding the results shown can someone define a solid definition for the standard deviation and the intuition behind it?
 A: 
A typical statistics course will define the standard deviation as "the average of the difference between the data set and the mean ".

That is false. I doubt that you've seen that in any textbook on probability or statistics.

So if we tried to describe the definition mathematically we should derive this equation (∑(|x-mean|))/n

That is NOT the standard deviation. That is the mean absolute deviation. It is quite intuitive, but it lacks a useful property: the variance (i.e. the square of the standard deviation) of the sum of independent random variables is the sum of their variances.
Both the standard deviation and the mean absolute deviation are measures of dispersion in that: (1) they don't change if one number is added to all of the numbers in the list and (2) if you multiply all of the numbers in the list by one number, then you multiply the measure of dispersion by the absolute value of that number.

while, the known law for standard deviation is (∑(|x-mean|))/sqrt(n)

No, it is not. Where did you find that? The square root of $n$ in the denominator shows up when you talk about the standard deviation of a sample mean, but there's nothing like that in a definition of standard deviation.

Example if I have a list of 2 numbers [14,6]


the arithmetic mean = 10


the average of distance of each value from the mean should = 4 , (14-4 = 10, 6+4 = 10)

Correct.

while the standard deviation law will calculate

It will give you $4.$

So, regarding the results shown can someone define a solid definition for the standard deviation and the intuition behind it?

It is the square root of the average of the square of the difference between the realized values and their average.
The reason it is done that way is that that makes the standard deviation a quantity that satisfies the points numbered (1) and (2) above while also having the "useful property" referred to above.
