I've recently started studying Mathematics and I am very confused about polynomials. What are they? How do you do operations with them? What are their applications, how can they be useful?

Related to this topic, could you explain the meaning of this sentence to me as if I was dumb?: "This theorem is guaranteed to work on a number n in about (log n)^12 steps, that is, a number of steps bounded by a twelfth degree polynomial in the length (in bits) of the input, n".