STI/Double Tap shifters
STI or Double Tap shifters are what most would consider to be traditional road bike style shifters. They use a shifter/brake combo that goes onto the downcurve of a road bike handlebar. Both SRAM and Shimano make lines of road bike shifters that are commonly seen on dedicated road bikes, gravel bikes and cyclocross bikes.
Flat bar shifters (trigger/grip)
Flat bar shifters are most commonly seen on mountain bikes, hybrids and cruiser bikes. They come in two different variations, grip shifters and trigger shifters. Grip shifters use a twisting motion to pull or release the cables to change gears. Trigger shifters use buttons/triggers to move between different gears.
Friction shifters are far less common than they originally were, although they are spotted occasionally. Friction shifters are usually mounted on the downtube of a bike and use friction to shift between gears; riders must feel the gears and adjust the shifters accordingly.