I was reading these notions from "Rational Homotopy Theory" by Felix, Halperin, and Thomas.
The notion of weak homotopy type is as follows: two spaces $X$ and $Y$ are said to be weak homotopy type if there is a zigzag of weak homotopy equivalences i.e., there is a sequence $$X\to Z_0 \leftarrow \cdots \to Z_k\leftarrow Y$$ with each arrow a weak homotopy equivalence.
The notion of weak equivalence for commutative cochain algebras is also similarly defined.
Motivated from above one may define the notion of rational homotopy type in a similar fashion. Two simply connected spaces $A$ and $B$ are said to be of rational homotopy type if there is a zigzag of rational homotopy equivalences i.e., there is a sequence $$A\to U_0 \leftarrow \cdots \to U_k\leftarrow B$$ with each arrow a rational homotopy equivalence (each arrow is a quasi-isomorphism).
I am not able to get a nontrivial example of two spaces having the same rational homotopy type. I want two simply connected spaces, with rational cohomology finite type, with no direct arrow that is a quasi-isomorphism but connected by a zigzag of rational homotopy equivalence.
