There are lots of fun characterizations of them. The most significant one is that they are the polynomials that minimize the uniform norm on a given set -- classically, this set was the closed interval [-1, 1] or sometimes [-2, 2]. This means they are the polynomials that attain the smallest maximum absolute value on the set or interval of interest. This minimax property is essential for their utility and ubiquity throughout pure and applied math

The minimax property is one of the ways you can generalize to other kinds of sets, allowing you to talk about the Chebyshev polynomial of, say, an arc or a Jordan curve or a union of intervals, and many of the properties enjoyed by the classical Chebyshev polynomials end up carrying over as well, but faaaaar less is known about these generalizations. In many cases all you can hope for are asymptotics, and you need much more in the way of machinery and sophisticated tools to prove anything -- even to compute explicit formulas for the coefficients.

The classical case is nice though because they can be explored fruitfully without very much background