The chain is the most important part of the "drivetrain" on any bike, be that on a single speed or geared bike. The chain connects the front and rear sprockets on the bike and provides the drive to the back wheel. A badly worn chain will slip and can be dangerous under high loads such as sprinting or riding up hill. The chain needs to be kept clean and there are many products on the market which will help with cleaning the chain on the bike without having to remove it. Chains for road bikes and mountain bikes are very similar and you would buy one to suit the number of gears your bike has. If you buy the wrong size chain, then you will risk the chain slipping and or damaging the other gear parts on your bike.
You may think that a chain is just a chain but branded chains from the likes of Shimano, KMC and SRAM have years of development. The pins and side plates are better quality than most and offer quieter gear changing and less slipping. This is due to tiny curves and chamfers built into the plates. SRAM make use of their Powerlink connector to link the chain together. This can be installed and removed without any tools. There are different widths and sizes of chain depending on how many gears you have on your bike and be aware that most chains will not work on a Campagnolo drivetrain unless specifically stated. Most cyclist that use Campagnolo will only use a Campagnolo chain as it is a slightly different size to the likes of Shimano and SRAM.