In the broad sense of the word, a topological string is a 2-dimensional TQFT. In its refined form this goes by the name TCFT. The “C” standing for conformal field theory points to what historically was the main inspiration and still is the default meaning of topological strings: the A-model and B-model 2d TQFTs, which are each obtained by a “topological twisting” of 2d SCFTs.
Accordingly, much of “physical” string theory has its analogs in topological string theory. Notably the toplogical analogs of the D-branes of the physical string – the A-branes and B-branes – have been studied in great (mathematical) detail, giving rise to homological mirror symmetry and, eventually, the notion of TCFT itself.
Also the perspective of string theory as the dimensional reduction of a conjectured UV-completion of 11-dimensional supergravity – “M-theory” – has its analog for topological strings, going, accordingly, by the term topological M-theory.
wikipedia topological string theory
Andrew Neitzke, Cumrun Vafa, Topological strings and their physical applications, hep-th/0410178
Mina Aganagić, Cumrun Vafa, Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots, arxiv/1204.4709
Marcel Vonk, A mini-course on topological strings (arXiv:hep-th/0504147)
The relation to topological M-theory/the topological membrane is discussed for instance in