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.
Front derailleurs can fit to the bike using either a band on clamp that is sometimes size specific or braze on. On many of the band on versions they come with a selection of shims so they fit a range of seat tube diameters however be aware that some only fit one size. Braze on bolt onto a bracket that is riveted to the frame when manufactured. These are nice and light as they don't require the band.
Most front derailleurs are 2 speed of 3 speed depending on how many chain rings you have on your chainset.
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.