Distance Between 2 Points
Here is how to calculate the distance between two points when you know their coordinates:
Let us call the two points A and B  
We can run lines down from A, and along from B, to make a Right Angled Triangle. And with a little help from Pythagoras we know that: a^{2} + b^{2} = c^{2} 

Now label the coordinates of points A and B. x_{A} means the xcoordinate of point A The horizontal distance "a" is (x_{A}  x_{B}) The vertical distance "b" is (y_{A}  y_{B}) 
So now we can solve for c (the distance between the points):
Start with:  c^{2} = a^{2} + b^{2}  
Put in the calculations for a and b:  c^{2} = (x_{A}  x_{B})^{2} + (y_{A}  y_{B})^{2}  
And the final result: 
Examples
Example 1

Example 2
It doesn't matter what order the points are in, because squaring removes any negatives:

Example 3
And here is another example with some negative coordinates ... it all still works:
(Note √136 can be further simplified to 2√34 if you want) 
Three or More Dimensions
It works perfectly well in 3 (or more dimensions) !
Square the difference for each axis, then sum them up and take the square root:
The distance between the two points (9,2,7) and (4,8,10) is:
