Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It has two approaches: bottom-up, which solves subproblems first before combining their solutions, and top-down (memoization), which caches results of functions to avoid repeating calculations. Examples covered include calculating the Fibonacci sequence, the 0-1 knapsack problem modeled with a dynamic programming matrix, and the coin change problem solved by considering all combinations of previous amounts.