The switching lever and/or levers, with which you "order" the gear changes also is part of a complete gear shift assembly on the MTB. Shifters should work precisely and smoothly. But the manufacturers, such as Shimano or SRAM, by now pay attention to a lot more details, such as e.g. ergonomic grip shapes, as little as possible weight and a clear optics at the cockpit.
What the Shimano switching lever or the shifter at the rotary handle (also called grip shift) are, is called the trigger with SRAM. MTB shifters are available with or without gear display and they have been coordinated to different rear derailleurs by the manufacturers. You have to choose the fitting shifter in accordance with your transmission at the MTB. When dealing with Shimano, it is simple to say that all 11-speed levers are also compatible with all Shimano 11-speed rear derailleurs on a mountain bike. The same applies to the 9 speed Shimano levers which are compatible with all 9 speed Shimano rear derailleurs. If you are dealing with SRAM, you should have a closer look since here you'll find that different compatibilities still exist. Some shifters are available in a set (pair), others are available separately for the left or the right side. Whoever is a fan of only one chain ring in the front, can do without the left-side switching lever. Here, you'll find the fitting shifters fitting to your rear derailleur for mountain bikes.