I will address this in two parts. Initially, I will touch on more ancient uses of what we now know as showers, and afterwards touch on a modern mainstream use of showers as we know them today.

As I'm sure you can imagine, the original shower was the natural waterfall, which have been used to clean and wash ones self since the beginning of human existence, essentially being a more efficient way of cleaning yourself than just jumping in a pool of clean water. Waterfalls were also popular because the water was normally clean because it had just run off of high ground and through minerals and the like, which is more reliable than pools of water which could be stagnant or infected in a multitude of ways.

The ancient Greeks, as usual, were the first civilisation to have showers as we know them, famously running clean water through lead pipes and pumping them out into communal shower rooms, which were used by members of all different classes. We can see proof of this on various found pieces of pottery from that time and, according to Wikipedia (I was not aware of this) "at the site of the city Pergamum". These shower rooms were as common as the famous bath houses of the time. They stopped being used so commonly when the Roman Empire fell. They are very comparable to modern shower/locker rooms and had places to store clothes, valuables, etc.

Regarding more modern uses of showers, they were first patented by a guy called William Feetham in 1767 in London. It is likely people had been using a similar system for a few years before he patented it, but it is effectively the shower you know now. There was basically a pump that pushed the (normally dirty) water into a pipe above your head that would be released by a chain when you wanted to actually shower yourself.

The modern shower became popular among elites when it was installed in French Military barracks in the late 1800s (I can remember when, sorry). The first public bathhouse was then opened in Vienna, Austria in 1887, later becoming mainstream and very popular in Paris just before the turn of the century. At this point, many of the European elite got showers installed, painting the pipes like vines or bamboo.

Showers didn't become a regular part of the household until the 1920s to 1950s, before which they were a status symbol or used for militaries for cleanliness. Your average guy would just visit the bathhouse if he wanted a shower, otherwise just using a basin.

Hope this helped.

In addition to the wonderful answer by the other poster, the episode of the askhistorians podcast entitled "cleanliness and hygiene in the early united states" from may of 2016 includes an interesting overview of this topic.