Big O Notation Explained
O(f(n)) means the family of function of f(n), a family of function for example could be {n^2, 3*n^2, 6*n^2, ...k*n^2}, for g(n) to be in the O(f(n)), it means that g(n) is less than all elements of f(n) in the long run. In the above example, it could be g(n) = n, versus O(f(n)) = O(n^2), obviously, n^2 is bigger than n for all n in the long run.

Absolute value sign compares the absolute values of these two functions so that even if g(n) is negative, we can compare which negative grows faster i.e. -n^2 grows faster than -n is equivalent to n^2 grows faster than n

n>N means in the long run

Bubble sort runs O(n^2) in the worst case means that Bubble sort can run much faster than O(n^2), but it does not exceed O(n^2). For example, we want to sort two elements {2,1} using bubble sort, obviously this takes 1 operation and that's it, 1 is smaller than n^2.