Derailleur gear systems have been around for over 50 years and were developed by Campagnolo in Italy. The system has stayed pretty much the same since its invention. The modern day derailleurs are a lot more refined but still based on the same design principle. The front and rear mechs of your geared bike do the work of chaining the gears and need to be kept aligned, cleaned and well lubricated to function correctly. With such items, the price variance is generally down to the more expensive ones being lighter, made of more complex materials and having better quality springs or bearings. The more precise the mechanism the better. The more expensive the derailleur the more precise its shifting will be and the likelihood of a missed shift is much reduced which is really important for competition riding.
Rear derailleurs come in various shapes and sizes. Some are mounted beneath the rear axle nut, most to a dropout which hangs from the frame behind the axle and a select few can bolt into the axle. The majority use the dropout fit. They range from 5 speed upto 11 speed and are made from steel, alloy or carbon. Alloy and carbon are your mid to high end components and the steel covers lower price models usually under £250.
The rear derailleur is controlled by the right hand shifter and this is the shifter you will use the most. Bike brands usually have the rear derailleur as the highest level component on the bike as it is the most used and how this part works and feels is at the heart of how your bike changes gear.
At the higher end of road bike manufacturing we are now seeing electronic gears which are made by Shimano, SRAM and Campagnolo, on these bikes the mechanisms are motorised and controlled by sophisticated circuits and wiring. The batteries last for approximately 1000 gear changes in-between charges.
The big three brands to look out for are Shimano, Campagnolo and Sram. These three brands made gear systems for all types of road bikes. Shimano and Sram dominate the mountain bike market. It's worth noting that gear systems from all of the above brands are not interchangeable and you have to run like branded components together except for chains and cassettes on occasion.