and
A dg-algebra $A$ is a formal dg-algebra if in the homotopy category of the model structure on dg-algebras it is isomorphic
to its chain (co)homology (regarded as a dg-algebra with trivial differential). Since all dg-algebras are fibrant in the standard model, this is equivalent to the existence of a span
of quasi-isomorphisms of dg-algebras.
In rational homotopy theory rational topological spaces are encoded in their dg-algebras of Sullivan forms. A simply connected topological space $X$ whose dg-algebra of Sullivan forms $\Omega^\bullet(X)$ is formal is called a formal topological space. (One can also say a formal rational space, to distinguish from the unrelated formal spaces in formal topology.) Such a space represents a formal homotopy type.
Examples are
compact Kähler manifolds (e.g. smooth projective varieties);
classifying spaces of Lie groups;
some homogeneous spaces $G/H$
the unstable Thom spaces $M U_n$ and $M S O_n$
the space $K(\mathbb{Z},2)$,
whose Sullivan minimal model is the dg-algebra on a single degree-2 generator with trivial differential.
For an early discussion of formal dg-algebras in the context of rational homotopy theory see section 12 of
Dennis Sullivan, Infinitesimal computations in topology , Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam: djvu, pdf)
Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274, doi
A survey is around definition 2.1 of
Review with an eye towards the dd^c-lemma and H-cohomology is in