rational expression
A rational expression is just a fraction whose top and bottom are polynomials instead of plain numbers. If you are comfortable with 3/4 — three parts out of four — then (x + 1)/(x - 2) is the same idea, except the parts are described with letters. The word “rational” comes from “ratio”: it is a ratio of two polynomials.
Formally, a rational expression has the form P/Q where P and Q are polynomials and Q is not the zero polynomial. The expression x^2/(x - 3), (2x + 5)/1, and 7/(x^2 + 1) are all rational expressions. A polynomial by itself counts too, since you can write it over a denominator of 1.
The one rule you can never break: the denominator may not equal zero, because dividing by zero is undefined. So every rational expression silently carries a list of forbidden inputs — the values of the variable that would make the bottom zero. These are studied carefully because they shape the expression's domain.
(x^2 - 1)/(x + 1) is a rational expression. Its numerator and denominator are both polynomials, and it is defined for every x except x = -1.
A polynomial over a polynomial — with one forbidden input.
Do not confuse a rational expression (a ratio of polynomials in a variable) with a rational number (a ratio of two integers). They share the name because both are “ratios,” but one lives in algebra and the other in arithmetic.