Yandex is the company that operates the Yandex.Taxi ride-hailing service.
Yandex.Taxi began as a Yandex service for booking taxi rides through a website and mobile app. Yandex is a Russian technology company best known for its search engine, but its wider ecosystem has included transportation, maps, navigation, food delivery and other digital services. Yandex.Taxi uses location technology, driver-partner networks and automated pricing and dispatching to match passengers with rides.
The service is also commonly associated with the name Yandex Go, a broader consumer app that combines transportation with additional local services in some markets. That branding can make the answer appear more complicated, but Yandex remains the parent company behind the Yandex.Taxi operation described in this question.
Another source of confusion is Uber. Yandex and Uber created a joint venture for ride-hailing operations in several markets in 2018, but Uber was not the company name traditionally used for Yandex.Taxi. Mail.ru and Didi are separate technology or mobility companies, not the operator identified by this service name.