If you are not a cyclist, any bike with drop handlebars is a "Road" bike and any bike with flat handlebars is a "Mountain" bike. If the person looks silly on the bike, it's probably a "BMX" bike. If the bike looks like an old Schwinn, it's a "Cruiser". If the bike looks like something you'd find in Europe, it's a "City Bike".
Bikes are far more varied than this, of course. Thus, once you enter the world of a person who wants a decent bike, you need to learn new words. I'm mostly going to talk about road bikes here, because that's what Joy and I have been talking about, and also, since I've already got a mountain bike, I don't need a second one.
Most of this is predicated on a simple thesis: If I, as a mildly overweight but ambitious cyclist, am comfortably dropping skinny guys who are on that style of bike, it is no better than a cross-country mountain bike with the tires swapped out.
Thus, there is a category of bikes you can buy, with largely road gearing and road parts, with a lightweight frame, and with wheels and tires slightly wider than road wheels and tires, that sometimes gets called a "Hybrid" but I tend to think is most properly called a "Flat bar road bike", that is completely without merit.
The big difference is that a hybrid will tend to have mountain bike components and a Flat-bar road bike will have road bike components. So, a "Hybrid" might have sturdy and decently wide wheels, 7-9 speeds over a fairly wide gearing range, V-brakes, and maybe a front suspension fork... a "Flat bar road bike" will have narrow lightweight wheels, a 9-10 speeds over a much narrower range, mini-V-brakes, and generally lighter-weight, less sturdy carbon fiber parts.
If you want a bike that's not going to be comfortable for hundreds of miles and isn't aerodynamic enough to reach a proper top speed, you might as well get something that's more useful. The added lower gears, improved off-road ability, and better resilience to road debris that a hybrid or even just a mountain bike with the knobby tires replaced by slicks offers make up for the disadvantages in ways that a flat-bar road bike can't.
The problem with a flat-bar road bike is that any design of bike with more than a few gearing selections can get to 15 mph comfortably without rolling resistance or anything slowing you down. Beyond 15mph, aerodynamics start to become a huge concern. If you ride 15mph for long enough, comfort starts to become a concern, too. Thus, what you really want is several different positions so that you can switch between them while riding. And you really want the aerodynamic position to be comfortable and have controls available.
In either case, a flat bar is far from optimal because you only have one position for your hands, maybe two if you add bar-ends. This works awfully well when you are off-road because that one position is going to give you the most control, plus the resulting bar is going to be both light and sturdy. But the second position doesn't generally have any controls to brake or shift. You can add a "mustache" semi-drop bar or a "butterfly" trekking bar, but that's still oriented towards a fairly upright bike.
Most people think that drop bars are inherently uncomfortable, but I suspect this is largely a matter of how you set up the front end of the bike. See, if you go into a bike store and try out a racing-oriented road bike of the correct size, the stem will be angled downwards, such that it's roughly parallel to the ground once you consider both the head tube angle and the stem angle. This is largely for historical reasons, but if you just flip the stem and potentially replace it with a shorter stem or a stem with a different angle, you'll get the handlebars in a more reasonable position.
The big advantage is that you get two good positions where you can reach the controls, one higher (the "hoods") and one lower (the "drops"). Plus, you can put your hands on the flat section, where there are optional secondary brake levers that can be mounted. Or you can put your hands on the very bottom section (the "hooks"). Thus, if you are comfortable with having your back flatter and thus your position more aerodynamic, you can have the bars low, such that even in the hoods, you'll have a flat back. Or, you can have it set up such that your hoods give you a unstressed angled back and you need to go into the drops more often.
The problem with pretty much all of the flat bar designs is that you don't get the controls in more than one position.
You can also get "aerobars" that clamp to handlebars and give you a stretched-out position with your arms and hands straight and in front of you. Or, you can get a bar that removes the "drops" and "hooks" position and has the aerobars integrated. The funny part abut the aerobar position is that a lot of long-distance riders swear by them because it's yet another position they can cycle through, whereas for time-trial events, the rider expects to spend as much time as possible in the aerobar position, regardless of comfort, because of the aerodynamic advantages.
There are a lot of "just because" things about mountain bikes. Just because the first people to make a "mountain bike" had old Cruisers, the rims are the same. Because they had to use motorcycle brakes instead of buying a beefed up bike brake, they tended to use flat handlebars. Grant Petersen, when he tweaked the prevailing mountain bike geometry to make it better for climbing while working at Bridgestone, made some mountain bikes with drop handlebars.
However, as time has gone on, it's nearly impossible to make a mountain bike with drops that still uses a fairly modern assortment of parts. The cable travel is different, so you have to use road brakes and shifters with road derailleurs and brakes and mountain brakes and shifters with mountain brakes and derailleurs.
Now, if you were to watch me on my mountain bike, you'll notice that I've got bar ends and that I use them a lot. Consider the geometry of a flat bar and longer bar ends compared to a roadie drop bar.
Both bars have a section between the stem and the bar ends to put your hands. This position works quite well for tricky manuvering on rough terrain. That's where all of the controls of a mountain bike are clustered. If you compare a road racing bike to a mountain bike, you'll be instantly annoyed because you can't brake or shift from your customary position on unstable ground because you won't have brakes or shifters in the flat position. However, if you add cross levers for brakes, you can brake from this position.
Now, if you add some bar ends, you've got a second position. As it turns out, this is fairly analogous to the "hoods" on a drop bar. If you grip the bar-ends on a mountain bar, you can get lower and more aerodynamic for a sprint, shift your center-of-gravity forwards so you don't endo while climbing, and use them like bullhorns on a sprint bike for the most optimal power-generating position.
Now, back in the early days of mountain bike history, Grant Petersen offered a few of them with drop bars. They didn't catch on, but I've got the distinct impression that he's right. See, if you compare the hoods on a drop bar to the bar ends, you realize that you can shift and brake in this position, which helps with a road descent and any sort of climb.
Now, you get two more positions. You can get lower on drops by grabbing the drops, which lets you get your center-of-gravity even lower and be both more aerodynamic and even less likely to endo, or you can even grab the hooks on the bottom.
Is it a disadvantage that you can't shift from the flat portion? I tend to think not. Most of the cases where you need to shift RIGHT NOW are covered by the other positions, once you start to use them.
There's also an application that's important for those of us who don't rely on somebody with a pickup truck to meet us at the end of the downhill and drop us off at the top of the hill again, which is to give your hands a second position on long rides over pavement. Drops are far superior in this case.
I suspect that, once you cross-over from cross-country into all-mountain and downhill, drops stop being quite as useful. Because if 90% of your ride is either uphill or downhill where you need that extra control, you might as well snip off the fancy ends and go back to a straight bar.
For the casual road rider, I also suspect that Grant Petersen is probably right that most folks would probably be happy with mustache bars or some of the variants on north road.
The problem, yet again, is that nasty "Just because" issue. Mountain bikers will never accept a bike with drops and roadies won't shame themselves into using mountain bike gearing. Everybody arbitrarily decided that drop bars were inherently uncomfortable without first looking at the front-end geometry and fit issues. And there's very little interest on the part of the groupo makers to make the mountain and road parts interchangeable because this allows them to charge more for road parts for no good reason.
Still, I much prefer my bike with bar-ends and a flat bar over a riser bar. I suspect that, for biking in traffic, bullhorns are about right. I've given some thought into what it would take for me to either make a rational drop-bar conversion or just set up my bike such that it has bullhorns with shifters and brakes on the end. Or even just a second set of brake levers. But, in the end, all of these tweaks require a certain degree of custom engineering that is probably more trouble than it's worth.