Welcome to the board. That can be a long topic but let me see if I can give you a quick version. Originally, single hose regs were the cheap alternative to the more expensive and better quality double hose regs. Early single hose regs were a good bit less expensive than the double hose regs and most of the early ones breathed horribly, even by the standards of the day. As they became more popular they also improved in design to rival the performance of the DHs of the day. The single hose has one major advantage (other than cost) over a DH reg, the second stage diaphram is located at the divers mouth and not behind them. Because of this, the breathing resistance of a single hose does not change with diver position as opposed to the DH which will change breathing resistance as the diver changes position. A correctly functioning DH reg will breath harder as you go into a head down postion, easier in a head up position and it will likely freeflow if you roll over on your back. As add ons like SPGs, octos and LP inflators became the rage it made more sence for the manufactures to drop the more expensive DH regs for the more profitable SH regs. With a single hose reg, adding more ports for these dodads is rarely more complex than drilling another hole in the reg body as opposed to redesigning the more expensive DH into an even more expensive reg.
Still, there are those of us who prefer the DH regs. I started on single hose regs and still dive them at times but ever since I started diving with an old DH that has been upgraded with modern internal parts and hoses it has become my reg of choice. Then there is the cool factor. No one ever walks up and ask about my Zeagle or single hose Aqualung reg but it's rare that I dive the old DH reg I don't get ask a lot of questions. If you get the oppertunity to dive a DH reg with someone who regularly dives them, give them a try, you may find you enjoy them.