The Simplex algorithm, developed by George Dantzig in 1947, is the most widely used method for solving linear programming problems. It operates by moving from one vertex (corner point) of the feasible region to an adjacent vertex, improving the objective function value at each step until an optimal solution is reached. The algorithm uses pivot operations on a tableau representation and typically terminates in polynomial time for most practical problems. Key steps include: selecting entering and leaving variables, performing pivot operations, and checking optimality conditions.