Why did we invent recumbent bikes in the first place? There are two main reasons: comfort and aerodynamics.
When we want to rest, we don't usually perch on a railing. It's much better to dive into an armchair and throw our feet on a table, isn't it? Now put some wheels under the chair, replace the table with pedals, add handlebars under your hands - and here you are.
Such position doesn't cause neck, crotch or wrist pain. The only potentially aching bit are buttocks which contain the main pedaling muscles, these can't work properly when we sit on them. But this is easily cured by laying the seat more back.
Since we left the trees and learnt to walk on our rear limbs, our most natural position is with our head up, feet down and face forward.
But the upright position is designed for walking or running speeds only. If we try to move faster than that, we get slowed down by our own personal headwind.
What if we change our position to better pierce the air? There are many examples in nature:
The fastest animals use headfirst horizontal position exclusively, but it doesn't work for humans that well. First problem is the view (straight down), second is the breathing (we lay on our lungs). It's better to leave natural examples behind and turn face up, with our legs first for practical and safety reasons. And here we go:
The laid back position is comfortable and aerodynamic, but is presents several technical challenges.
Nobody has yet invented any miraculous mechanism that would beat chain transmission in all aspects, so we keep with the chain. Bad news is that 'bents have cranks all the way forward and rear wheel is far far away.
One way is to use idlers and route the chain around the front wheel and seat, the other way is to build the bike around an ideally straight chain line (or with the front wheel in front of the cranks). First solution is better for low, streamlined racers that profit from better aerodynamics more than they lose in transmission efficiency and complexity. Touring bikes usually use the more simple and reliable second solution, or some compromise between the two.
But there is also another way. Front wheel doesn't have to be just an obstacle for the chain - it can be the destination as well. Unless we want to ride mountain trails where front wheel drive slips, of course. We save half of the chain length and don't have to worry about seat position. But we do need to worry about the steering (an obvious solution would be to drive front wheel and steer rear one, but nobody has yet figured out how to make such a bike reliably stable and controllable). We can either put the cranks on the main frame and let the chain twist in turns, or put the cranks on the front fork and let our legs twist.
The first approach is easier to control; turning radius is somehow limited, but it's not a problem most of the time. Racing 'bents are often built this way. The second design requires the legs to pedal and steer at the same time and it has either limited turning radius (because knees can't bend sideways), bad static stability (wheel flop) or no dynamic stability at all (because of negative trail).
A general disadvantage of horizontal pedaling is that the feet need to be kept on the pedals by force, not by gravity. That means constant pushing, even on the return stroke where it robs a lot of power. Fortunately, clipless pedals (SPD etc.) solve this problem completely.
On an upright bike, our hands are directly over the front wheel, just add handlebars. On a 'bent, the hands are further back. How to connect them with the front wheel? There are generally two ways. Either mount the handlebars directly to the stem and get a long "tiller", or place the handlebars on a separate bearing and connect it with the fork by some transmission (a simple rod is most popular). The first way is easier to build, but its turning radius is limited by how far we can push the tiller. And we must get used to seemingly reversed steering, but that's usually not a problem. Remote steering is more ergonomic and allows tighter turns and adjustable sensitivity and position, but it presents several more parts, joints and bearings that can break down.
Our next choice is where to put the handlebars: either under the seat, or above. Underseat steering is more comfortable, hands rest on the handles like on a couch. You can also pull on the bars to prevent being knocked off the seat in bumpy terrain. On the other hand, the handlebars need to be rather wide which is neither aerodynamic nor easy to push through doors. Overseat handlebars can be narrower and therefore more aero and easier to walk. In addition, they give you place for lights, speedometers, maps, bags and so on. Disadvantages are the higher hand position is less comfortable and that you have the steering column right between your legs, which is not what you want in case of a crash.
Upright bikes have an advantage in rough terrain: we can stand up, use our legs as springs and let the bike bounce under us. On a 'bent, we lay like a potato bag and get bounced around by everything the bike doesn't absorb. That's why so many recumbents are suspended.
From a design point of view, it's usually the same as with a standard bike: telescopic front fork and a swingarm at the rear.
Recumbents are becoming less and less rare, but there are still many rumors and false beliefs about them. Let's shed some light here.