Microtransit

Microtransit is a form of demand responsive transport. This transit service offers a highly flexible routing and/or highly flexible scheduling of minibus vehicles shared with other passengers. Microtransit providers build routes ad-hoc exclusively so as to only match each demand (trip) and supply (driven vehicle) and extend the efficiency and accessibility of the transit service. Possible pick-up/drop-off stops are restricted (usually within a geofenced area), and transit can be provided either as a stop-to-stop service or curb-to-curb service. 
Proponents argue conceptually, microtransit fits somewhere between private individual transportation (cars or taxicabs or ridesharing companies) and public mass transit (bus).
History.
Although the share-taxi kind of transit service has been running for a while in Southern hemisphere countries and Asia, these have involved private provision of some degree of fixed routes or fixed schedules and not always booking ability let alone mobile booking or route optimisation. The development of mobile booking technologies has led to a wave of pilot schemes and adoption in Europe and North America.
Development.
Technologies allow real time exchange of booking information and programmed route optimization of the transit service. The term Microtransit may have emerged into widespread industry discussion around 2015, when this wave of technology-enabled services was starting, and seems specific to the English language.
The current implementations result from public-private partnerships (and subsidized by the government) or are brought by the private sector directly to the customer. It is unsure if microtransit can be profitable (just like public transit).
Success of microtransit systems depends on its configuration. Some experiences in the US resulted in failures.
Application.
The flexibility and intelligence in microtransit can be useful in cases when the demand is either geographically spread or coming at various and/or unpredictable times, i.e. when it is hard to gather demand with a planned transit service. Examples include: low-density areas, night services, and other formats adapted to specific needs.