I'll take a crack at this, I'm not a historian but I do have a UCI bicycle racing license (lol) and know a pretty good amount of the technical aspects of bike history. I'll give a brief overview, noting on some of the major developments.
The earliest bicycle designs were the velocipede or dandy horse. This was something that superficially looks like a modern bicycle (two equal sized wheels), but did not yet have pedals or a drivetrain - you operated it by pushing off with your feet.
In the latter half of the 1800s, pedals became common, but drivetrain technology was not there yet. These bicycles had direct pedal drive of the front wheel. Eventually these designs matured into the Penny Farthing, which is what most people would think of as an old-timey bike. It's called such because the two different wheel sizes resemble a penny coin and a farthing coin next to each other.
As an aside, let me describe something called gear inches. This is a term used by bike engineers to describe the equivalent penny-farthing wheel diameter. A penny-farthing with a 60" diameter wheel will move 188" (60" times pi) with each full rotation of the pedals. For a typical experienced rider, 90 rpm is a normal cadence, so a 60" penny farthing at 90rpm goes 16 mph. To go faster, you can increase the cadence, although even the pros have a hard time holding >120 rpm. Or, you can make the wheel larger diameter. The size of the front wheel of a penny farthing was limited by the length of the riders legs, and as it got bigger, mounting and dismounting the bicycle became more challenging, as you are suspended higher in the air, and have to jump on and off of it while it's moving. For reference, a modern 2x11 speed bike as raced by the pros will have a lowest gear ratio of 32-36 gear-inches, and a highest gear ratio of 125-135 gear-inches. You'd need to be 10 feet tall to pedal a penny-farthing with that big of a wheel, but a modern bike can turn a ~27" wheel five times with one pedal stroke.
This led to the development of the Safety Bicycle, which is the first bicycle that really looks modern. This bike went back to having two equal sized wheels. Instead of the pedals being connected directly to the hub of the front wheel, they were now connected to a gear between the wheels, which then used a chain to drive another gear on the rear wheel. This brought the rider much closer to the ground for improved handling, balance and safety (you don't have to jump down from a couple feet off the ground any more). By using different size gears on the cranks and on the wheel, you could have a bike with a high gear ratio without having to put in a gigantic wheel, because each pedal stroke could be multiple revolutions of the wheel.
The next big invention was the pneumatic tire in 1888. By using a air-filled rubber tire, bikes could absorb bumps in the road much better.
By the mid 1890s, bicycles looked remarkable similar to those today - double diamond frame with two equal sized wheels, pneumatic tires, chain drive, etc. Some of the bike races that still occur today originated in that decade, such as the famous Paris-Roubaix
I'll add that at this point, bicycles were quite mainstream, with organized groups such as the League of American Wheelmen, and there is quite a bit of info out there about the popularity of bicycles, especially among women, but I think that might be better for a true historian to discuss.
The 1900s saw more gradual development of bicycles. Multiple gear ratios became more common beginning with the introduction of the Simplex derailleur in 1928, with various ways of actuating them. One of my favorites are suicide shifters, where one lever was used to loosen the rear wheel on the frame, and another was used to shift the chain onto a different cog, then the wheel was tightened again. This was on a 4-speed bike, in the 1940s.
By 1960 or so, road bikes at least were fully 'modern'. Derailleurs had been developed to allow shifting from a convenient paddle on the downtube of the bike without having to loosen a wheel or anything like that. This is a racing bike from 1960, and you can see it looks extremely similar to its 1990 equivalent
The 1970s in the US saw a "bike boom", where road bikes increased in popularity, but again I will leave it to a historian to discuss the cultural significance of this, I am trying to keep things to the technical aspects that I'm familiar with.
In the past 60 or so years, the majority of road bike development was moving to lighter materials (steel to aluminum, aluminum to carbon fiber), more gears (a 1960 bike might have had 5 cogs on the back, a modern bike has 11), fancier shifters (downtube shifters were replaced with integrated brake/shifter levers in the 1990s, and now electronic shifters are becoming popular).
However, in the 1980s in Marin County, California, the mountain biking movement began, first with homemade bikes converted from clunky beach cruiser bikes, but then eventually into a new category of bikes that went on to gain worldwide popularity. During the 1990s, there was a mountain bike boom.
To summarize