Polynomials are algebraic expressions with variables like x and y. They involve various powers of the variable that are multiplied by specific numbers and added or subtracted, along with the potential addition or subtraction of a number. The highest power used determines the degree of the polynomial, such as a first degree polynomial involving the highest power of x being 1, a second degree polynomial having the highest power of x as 2, and so on. Examples are provided of polynomials up to the third degree.