Sounds like an indexed thumb shifter or an indexed pod shifter or a trigger shifter. All three names mean the same kind of thing, though different designs may put the two levers in different places, or may use a button for release.
Shimano Tourney 7 speed TX50
Shimano Altus three position left hand shifter.
Its possible to get shifter/brake levers integrated into one unit. Advantage is that you have fewer mounting bands on your bars, so its a space saver and a small weight saving. Disadvantage is that changing a shifter MAY mean changing the brake lever too.
Shimano has a bunch of standards for combination shifter and brake levers, named I-Spec and available in variants called "I-Spec A" which is the original, and "I-Spec B" and "I-Spec II" none of which are inter-compatible.
There is nothing wrong with running a separate gear shifter and brake lever, as long as they're both reachable from your grips. From a future-proofing point of view, separate makes a lot of sense.
The grip shift, or revoshift, or many other names. These are commonly found on cheap nasty bikes known as BSOs, and are renowned for having a short life. They tend to crack rending them useless. The exception would be the Rohloff grip shifter which is super robust.
bar-end shifters - popular with touring riders, and can often run in indexed or friction modes. I find the cable routing looks weird, so its not uncommon to route the cable under the bartape, or even completely internal to the bars.
This is a friction shifter, where the position is determined by your skill only.
This is a pair of downtube shifters, more common on 80s road bikes. These never really appeared on MTB. Can be mounted on the stem, in which case they're called stem shifters.
Finally brifters - the gear change of choice of the drop bar rider. Common on Road bikes and Cyclocross, and gravel.