Valve lifters play a key role in the valvetrain of pushrod engines. They go all the way back to the earliest days of the internal combustion engine. The earliest engines did not have pushrods or rocker arms. They were a "flathead" design with the valves in the block. The lifters (also called "tappets" because of the clattering noise they produced) rode on the cam lobes in the block and actuated the valves directly. It was a simple design but not the best configuration for breathing efficiency or horsepower.
Relocating the valves to the cylinder head was a major step forward in "overhead valve" (OHV) engines because it allowed the engine to breathe more efficiently and develop more horsepower from the same displacement. The overhead valve design added complexity to the valvetrain because it required the addition of pushrods and rocker arms.
The lifters also had to route oil up through the pushrods to lubricate the upper valvetrain components.
With "overhead cam" (OHC) engines, the camshafts are in the cylinder head(s) and actuate the valves directly or via cam followers, so there are no lifters. However, most modern OHC engines do have some type of hydraulic valve lash adjusters. The adjuster may be mounted in the head and serve as a fulcrum point to maintain zero lash between the cam follower and valve, or located inside a bucket that fits over the top of the valve, or a "mini-adjuster" mounted in the end of the rocker arm.
