An Espresso Timeline

Angelo Moriondo of Turin patented his first espresso machine under the title of "New steam machinery for the economic and instantaneous confection of coffee beverage, method 'A. Moriondo'. The patent was updated on 20th November 1884, Vol 34, No, 381. The invention was then covered by International Patent after being registered in Paris on the 23rd of October 1885.
Luigi Bezzera files a patent for machine that contained a boiler and four "groups". Each group could take varying sized filters that contained the coffee. Boiling water and steam were forced through the coffee and into the cup. Ambrogio Fumagelli (see ref. later) claims that this was the birth of espresso coffee.
Bezzera's patent purchased by Desiderio Pavoni.
The Pavoni company begins manufacturing machines based on the Bezzera patent.
First espresso machine installed in the USA. A "La Pavoni" machine at Reggio's in New York. (Still there on display)
Earlier espresso machines forced steam through the coffee, causing a burnt flavour. In 1938 Cremonesi developed a piston pump that forced hot (but not boiling) water through the coffee. It is first installed at Achille Gaggia's coffee bar but World War II prevented further development at that time.
Gaggia begins manufacturing a commercial piston machine. The resulting coffee has a layer of foam or crema.

The Coffee Museum in London has several interesting machines on display, including this one:
Unfortunately, the museum has no information at all(!) about the machines on display but this machine may well be one of the early Gaggia machines that use a separate heat exchanger to avoid forcing steam through the coffee. (More photos from the museum below.)

Faema launches a pump based machine. Instead of a hand operated piston the water is forced through the coffee by an electric pump. Water is taken from the fresh water supply and travels through a tube that is passed through the boiler and then through the coffee. This allows the water to be at the optimal temperature (~200F), filtered and not have to stay in the boiler for a long period. Almost all modern restaurant machines are essentially this design.

Modern innovations continue in the design of espresso machines. Many of these innovations are designed to produce a consistent product irrespective of the operator.

Some changes are simple, such as precise metering of the amount of water passed through the coffee. Earlier designs were based on the operator deciding when to stop the flow.

Other developments include "fully automatic" machines that grind the beans, froth the milk and deliver a complete cup, all with the touch of a button.

Most of this information (but not the form of words) was obtained from various books, including "Coffeemakers" by Ambrogio Fumagalli. This is a fascinating book of photos of coffee machines from 1800 to 1960 with a little bit of history thrown in. The English version is published by Chronicle Books of San Francisco. The definitive work in the area is "Coffee Floats Tea Sinks" by Ian Bersten.

Wikipedia entries for "Espresso" and "Espresso Machine" contain more information.
A machine of similar style to the one in Regio's New York. (London Coffee Museum)
An early single group Faema hand pump machine. (London Coffee Museum)