The formal definition, according to Wikipedia, is "an expression consisting of variables and coefficients which only employs the operations of addition, subtraction, multiplication, and non-negative integer exponents."
Ok, that probably didn't help, did it? What does it really mean? Well, a polynomial is something that looks like this:$$ x^2+7x-9 $$
It contains variables (\(x\)), it uses multiplication, subtraction, and addition, and has a positive, integer exponent (the \(x^2\) term).