Understanding the complexity of an algorithm helps us decide whether or not we should use it as the design of a program to solve a problem. Complexity is usually measured in terms of the average number of steps in the computation of a program. The steps can be used to estimate an average bound, lower bound, and upper bound of the amount of time and for the amount of storage space needed for the computation. The lecture explains Big O notation and concept and, using recurrence relations, develops the Big O value for several types of computations. The steps of interest are the primitive steps of an algorithm and the operations that are intrinsic to the data structure used in the program implementation of the algorithm.
Big O estimates are usually associated with the algorithm -- its primitive operations and data structure. Algorithms are grouped into classes associated with linear, n log n, quadratic, and exponential Big O bounds: O(n), O(n times log n), O(n**2), and O(2**n), respectively. For a given value of n, the lecturer gives the values of n, n x log n, n**2, and 2**n, to show the dramatic growth as n grows. If a problems grows by a factor of n an algorithm that grows too fast, e.g. has quadratic or exponential growth, would not be a good choice for the design of a program to solve it.
Note that the details of complexity analysis require a strong mathematical background, and you should focus on the main ideas or 'big picture' first before delving into the details.