In looking at the problem of how dragons fly, it's worth starting with the simplest issue: why is it that no large flying animals exist? Large passenger or cargo planes can reach several hundred tons, whereas the heaviest bird we know of (Argentavis) was maybe 150 lb, and the largest flying creatures (Quetzalcoatlus, etc) might have weighed 500 lb (huge range of proposed values and ability to fly).
There are a couple of reasons for this, but the big one is muscle. To fly like a bird, you need muscles that are strong enough to move the wing, and that can run at a power level sufficient to fly. Compared to the stiff struts available to fixed wing aircraft, muscle just isn't very strong. Compared to jets, turbines, and other aircraft engines, muscle produces very little power for its weight.
Let's start by putting together our sample dragon. We want something big enough for a knight to ride on, so we'll give it a head and neck 5' long, a body 10' long and 3' thick, and a 15' tail. Total body volume depends on taper ratios, but probably a bit over 3 cubic yards, so at human density maybe 2.5 tons. We can probably justify bringing that down to about 1 ton. Now, in order to have a wing that looks reasonable folded next to the body, the fingers should be only slightly longer than the body, and the other two joints have a combined length similar to the fingers, so the total wingspan of our dragon should probably be not much above 40' with an aspect ratio of around 8 (200 square feet wing area). We'll give our dragon a drag coefficient of 0.025 (5 square feet), which would be reasonable to good for a modern aircraft; its best L/D ratio is 15.
Wing aerodynamics is a pain, but you can come up with plausible numbers by treating them as a form of helicopter (you can also treat them as a fixed wing plane, but flapping wings don't really have a stall speed as such, and at low speeds standard aircraft assumptions wind up off). Note that hovering or near-hovering flight has some wing design issues that may prevent it even if adequate power is available. In any case, over a speed of 0 to 100 mph, dragon power requirements are about as follows:
Optimal loiter speed (least energy/time) is 55 mph; optimal travel speed (least energy/distance) is 70 mph. Terminal velocity in a dive is 450 mph.
Now, consider the same dragon carrying a load. There are two reasonable options: a princess (120 lb, but lots of drag, call it 5 square feet) and a knight (250 lb, but with a proper stance and saddle maybe only 2 square feet drag area). Our power curves now look as follows:
As we can see, the weight severely affects low speed performance, the drag is more relevant to high speed performance (if the dragon wishes to carry the princess a long distance, he should probably eliminate any high drag clothing and hold her closely against his body).
This is a lot of power. We pretty much need our dragon to be able to manage 22 kW for a fairly extended period, and it can't efficiently carry a knight without at least 25 kW extended power output. For comparison, most horses cannot even sustain 1 horsepower (0.745 kW), and this dragon is only slightly larger than a draft horse, so our required sustained power output is 30-40 times greater than a horse (power density drops as size increases; weight for weight, the dragon only has around 10x the power output of a human professional athlete). In any case, however, this is dramatically in excess of any plausible power output for anything like an animal metabolism.
Now, let's look at the other problem: material strength. The basic strength requirement for the wing muscle is equal to (torque on wing) / (distance from wing joint to wing tendon). The strength of muscle varies a bit but generally won't go above 50 psi or so, and the torque on the wings also varies but probably peaks at around 40-50,000 ft-lb, so we want 1,000+ square inches of muscle, and given the general shape of the body we'll have trouble getting more than 100 square inches, so we really need something about ten times stronger than muscle.
The above dragon is fairly modest in size, of course. It's not small, but by legendary standards it isn't all that big either. If you increase size, keeping all proportions the same, mass increases as the 3rd power of size, flight speed increases as the 0.5 power, power requirement as the 3.5 power, material strength requirements as the 1st power.
Before considering how a dragon might power its movement, it's worth considering the topic of food. At typical muscle efficiencies, one kWh of power uses up about 5,000 Calories. Thus, an hour of flight will probably use up more than 120,000 Calories. Muscle also has a substantial resting upkeep cost, so daily power requirements are likely on the order of 250,000 Calories. An animal or person is probably worth around 500 Calories/lb, so we need 500 lb of prey per day, or 250 lb of prey to power an hour's worth of flight. That seems like an impractical amount of food to actually capture in one hour, and would certainly result in a single dragon absolutely ravaging the countryside, so we probably have to figure that the dragon gets most of its calories from somewhere else. We do know that dragons eat meat, but that might be for some reason other than power production.
The easiest way to speed up chemical reactions is heat. Given that dragons are associated with fire, and often are assumed to have internal flames of some sort, this seems consistent anyway. At this point, we're well outside of what biology does, but certainly machines can do all of this. The obvious first choice is wood, but wood-burning engines have low power density, so we probably need to do some fuel processing. The most likely model seems to be biomass gasification followed by something like Fischer-Tropsch synthesis, and to speed up the process the dragon might want to dig firepits and produce charcoal. This has the additional benefit that the area a dragon lives in will be fairly polluted by smoke and noxious fuel byproducts. In addition, this synthesis process produces a mixture of volatile gases (which would be useful for powering the dragon's metabolism) and tarry or waxy liquids (which aren't very useful as fuel, but with just a little more processing turns into the equivalent of napalm). In addition, since dragons would really prefer dry wood to green, it gives them a reason to raid villages (it's not that it wants to eat the people; it wants to eat the buildings). At typical combustion engine efficiencies one kWh represents slightly under a pound of fuel.
Well, we now have a quite suitable fuel for a modern airplane, but there really aren't any good hydrocarbon-burning muscle-type actuators, so we need to figure out a way to convert our fuel into something that can power an actuator, probably some sort of centralized engine. Doing this also helps deal with one of the oddities of dragon design: birds are pretty much all wing muscle, but the wing muscles on a dragon are actually pretty small. The difference can be explained if you have artificial muscles with a very high power density, but which require a large power plant of some sort to run them.
Given known modern technologies, the major candidates for this sort of artificial muscles are dielectric elastomers and pneumatic artificial muscles; both can achieve the required strength and power densities and have contractile behavior comparable to human muscle. In the first case, the dragon would have an electrical generator in its torso; in the second case, an air compressor (or other source of high pressure gas; with high temperature materials, steam might be an option). Both options can be powered readily by any of a variety of combustion engines, and the required power is actually pretty modest by standards of such engines; your average motorcycle engine has plenty of power. Of these options I tentatively prefer the pneumatic muscles, as it results in dragons that are naturally quite light for their volume, and the fact it already has an air compressor makes it easier to construct a flamethrower. However, 'lightning dragons' that work the other way seems plausible.
In biological terms, it's rather hard to figure some of this out. The fiber sleeves used for the muscles are well within what's plausible biologically, but the valve performance required is very high, and most certainly a combustion engine cannot be living tissue. The most plausible idea I can think of is some form of hard biological ceramic that becomes refractory when dry, which is used to form the engine (tooth enamel is the closest equivalent). Alternately, the dragon may have limited electrical capabilities. Whatever trick is used is probably also used for other parts of the dragon, such as its scales and bones (while the bone problem is less severe than the muscle problem, conventional bones still aren't really strong enough).
Okay, we know how strong a dragon's wings have to be: strong enough to carry the dragon. How strong are its other limbs? We can look at this in one of two ways: either they have to be strong enough to hold up the dragon, or they're whatever strength is appropriate given their dimensions.
If the dragon's front limbs are twice as long as human arms, and have similar proportions, they'll be four times stronger. However, we've established that dragon muscle is some ten times stronger, so they'd actually be forty times stronger than human limbs. Given that the dragon is only 13x the weight of an average human, that seems like slight overkill, but given that the front limbs are actually used as legs, whereas human forearms are not (and get tired rapidly if used for crawling) somewhat stronger is appropriate; we'll give the forelimbs 25x human ST, or ST 50; this implies a limb that is moderately thin. Now, the quickness of a limb (how long it takes to perform a specific action) scales with sqrt(strength/(length*weight)); a limb that's 2x longer would normally weight 8x as much, but we've already given our dragon 40% of normal mass, and the limb is slightly thinner, so it actually only weighs around 3x as much as a human arm, and will be about twice as quick -- meaning a dragon claw swipe takes half as long as a human punch, and moves four times as fast, but doesn't have an awful lot of mass behind it. Still, this seems plausible enough for a striking ST of 50. They are also subject to a property that would be weird for any other creature: they may get weaker when power is being used by other muscles (such as the wings).
Other parts of draconic anatomy are likely to be similar -- not very heavy (the tail may be almost totally hollow) but terrifyingly fast.
So, we have a candidate system for building a dragon. There are a wide variety of problems making these dragons biologically highly improbable, but they don't seem to be totally impossible. What are the consequences?
A typical dragon, then, smells strongly of wood smoke, and probably emits enough carbon monoxide to be hazardous. It may also emit a variety of other noxious substances as part of fuel processing. A dragon's home is likely to be surrounded by burnt things, often partially buried. It is at least believable that a dragon has uses for metals (probably as catalysts). A dragon's torso contains some high temperature components; if you stab the wrong spot you might encounter high temperature fluids or gases. A dragon's breath weapon is a spray of napalmed oil, propelled by high pressure air and ignited by some sort of chemical or spark; a dragon might have several gallons of this stuff stored up. A dragon probably can't fly very many hours per day, as it's limited by the rate at which it can synthesize fuel, though its metabolism might be able to convert some alcohols and oils more directly into fuel.
Like all flying creatures, dragons will be rather weight sensitive, and so their skins, while possibly quite tough, will not be thick. A pierced dragon will tend to produce air rather than blood, though the requirements of using them as muscles mean they will have valves that can cut off air flow quite quickly.