What makes running so much less energy-efficient than bicycling? Most people can ride 10 km on their bike. However, running 10 km is a lot harder to do. Why?
According to the law of conservation of energy, bicycling should be more intensive because you have to move a higher mass, requiring more kinetic energy to reach a certain speed. But the opposite is true. 
So, to fulfill this law, running must generate more heat. Why does it?
Some things I can think of as (partial) answers:


*

*You use more muscles to run.

*While running, you have more friction with the ground; continuously pouncing it dissipates energy to it.

*While you move your body at a slow speed, you need to move your arms and legs alternately at higher and lower speeds.

 A: *

*Running requires intense muscle contractions with a low duty cycle, whereas cycling uses long, smooth contractions. If you work at running, you can easily get to the point where it is no longer an aerobic challenge, yet still tough: you hardly have to breathe, yet the legs struggle with lactic acid buildup.

*Running wastes a lot more energy on compensating motions: the runner has to move her torso and arms to compensate the kicking motion of the legs.  The leg motion is not symmetric in running: the forward kick of the recovering leg is faster than the backward movement of the drive leg, and so the arm on the same side as the recovering leg has to swing backward to compensate its motion.  Compensation on a bike is mostly limited to a side-to-side rocking during a hard effort.   

*The runner basically flies through the air, but periodically comes down and touches the ground with one foot, and then exerts a force in order to get back up into the air. However, this is not done by an efficient, elastic bounce. The landing energy is dissipated, rather than stored and reused. In fact, the runner must exert energy in order to absorb the landing, and then exert more energy to get back into the air. The runner thus wastes considerable energy to stay in the air.

*Depending on the nature of the footstrike, the runner may counter-productively be exerting energy which retards (brake) his forward motion, and then has to exert more energy to recover momentum.
A: One word: inertia. When you're riding a bike on a level gradient you just need to give it a push to get going, then you can coast for quite a while before friction and air resistance slow you down. In other words, the relatively frictionless wheels mean the bicycle's kinetic energy doesn't dissipate quickly. But the human body doesn't have wheels, so while running you have to give a good kick to get going, and then another kick to keep going on the next step, and so on. When hills are involved the difference is even more pronounced, since we run downhill the same way we do on the level, by continually pushing ourselves forward; whereas on a bicycle you can take advantage of the slope and just coast down it.
I suspect that raising and lowering your centre of mass isn't as inefficient as the other answers have suggested. This is because your legs are springy, so at least to some extent you're just converting energy back and forth between gravitational potential and the spring force in your legs. Humans are possibly the most efficient long-distance runners in the animal kingdom. There is a school of thought that says the reason we are bipeds is that we evolved as endurance hunters, chasing our prey until it collapsed from exhaustion rather than trying to outrun it over short distances. Whether that's true or not, we probably wouldn't do all that bouncing up and down if there wasn't a good reason for it.
You might ask why, if using wheels is so much more efficient, didn't we evolve that instead? I don't know, but it seems no animal has been able to evolve wheeled locomotion.
A: A lot of the answers here go into movement of your center of gravity etc. I think it's a lot simpler than that. 
When you are cycling your vertical movement on the pedals is translated into horizontal movement of the wheels, coupled with inertia a small amount of energy can go a long way. 
Whereas whilst running you are putting energy into moving horizontally, to get places, as well as vertically to reduce friction with the floor. However all the vertical movement is fighting gravity, and wasted as your vertical movement gets you no closer to your destination
This answer assumes that the distance is over a flat surface. As soon as you throw a three dimensional aspect on it then the system turns on its head. 
For horizontal movement the bike is best as its wheels are designed to reduce friction, unlike feet. 
However as soon as you come to an incline the lack of friction will cause the bike to roll backwards unless constant energy is put in to the system, whereas the increased friction one has from running allows anyone to stop and stay put. 
As soon as you get to an incline from a standing start, running is much more efficient than cycling.
A: This is easiest to understand if you start by considering the extreme cases of steep uphill and downhill slopes.
Actually it isn't true in all cases that running is less efficient than cycling. In work by Minetti et al. ("Energy cost of walking and running at extreme
uphill and downhill slopes," DOI 10.1152/japplphysiol.01177.2001.), it was found that when elite mountain runners run uphill on a treadmill, at slopes greater than about 0.20, the efficiency becomes about 0.25, which is the efficiency of concentric muscle contractions. ("Concentric" means that the direction of motion is in the same direction as the contraction of the muscle, as when you do a pull-up.) This is an upper limit on the efficiency of any human-powered mode of going uphill, so since trained runners achieve it, they are not less efficient than cyclists on these slopes.
On a steep downhill, a cyclist can coast while the leg muscles expend zero energy. Running downhill does consume energy. In fact, the runner's efficiency is negative, because the gravitational potential energy of the person's body decreases, while the body's energy reserves are depleted. Minetti measured the downhill efficiency on grades steeper than $-0.20$ to be $-1.20$, and this is approximately the efficiency of muscles in eccentric contraction (like letting yourself down from a pull-up).
So if we can understand why lowering yourself down from a pull-up expends energy, then we automatically also have a physiological explanation of the imperfect efficiency of downhill running on the steepest grades, and then by interpolation we have an explanation of why running is less efficient than cycling on ordinary slopes or on the flats.
The reason that muscle tissue expends energy in eccentric contractions is that there are processes in the body that dissipate heat when a muscle is under tension. For example, the body has to burn fuel to maintain muscle tension, and there is also internal friction in the muscle as the muscle moves.
In addition to the processes described above, there are other energy-dissipating mechanisms as well, including the dissipation of energy into vibration and sound on a runner's foot strike.
A: 
Many of  us have ridden  bicycles at
  some time in our lives. and in fact
   this mode of transportation has become
  markedly more popular recently as a result of the energy
  shortage.  Each morning at  my own
  university, Duke, people can be
  seen riding machines with masses
   of $10$ to $20$  kilograms and struggling
  to  reach one  of the  campus entrances
  at the top of a long,  steep hill.
  As in  many other aspects of animal
   locomotion, there is a paradox here.
  Why should people  encumber
  themselves with such  heavy apparatus,
  particularly while  going uphill?
  Ask  a rider this  question, and
  the response is usually: "It's easier
  than walking" or "It's faster than
  walking." But  why should it be?
A number of incorrect  explanations
  are offered: "A bicycle  has gears."
  Shifting  gears allows the rider to
  vary the  speed at which the feet
  move; but even if  the foot speeds  of
  a cyclist and a pedestrian are
   matched, the cyclist still goes farther and in  less time on a  given
  amount of  energy than the pedestrian.
  "Your  weight is supported by
  the seat." But if you pedal  standing
  up, biking still is faster and less
   costly of energy than going  on foot.
  "Your center of gravity doesn't go
  up and down." But it  does if you
  pedal standing up. Why, then, is
  bicycling easier than walking  or
  running?
[…]*
We can now appreciate why bicycle
  riders are willing to propel  the
  extra weight of a bicycle, even
  when going uphill. The cost of
  transport on a bicycle is low because active muscles are  not
  stretched while  pedaling, and mean
  muscle  efficiency is about $.25$, nearly
  its maximum value. The wheels
  stabilize the rider's  center of mass.
  Even if the rider  accelerates the
  center of mass vertically by  pedaling while standing up, active muscles
  need not be stretched.  When
  the center of mass falls, the cranks,
  sprockets, chain, and rear wheel
  constitute a system of  levers that
  transposes the vertical motion to a
  horizontal  one by supplying a perpendicular
  force.  Thus, humans can
  use external machinery to move
   along a level surface with the same
  muscular efficiencies that swimming and flying animals achieve
  naturally.

The Energetic Cost of Moving About: Walking and running are extremely inefficient forms of locomotion. Much greater efficiency is achieved by birds, fish—and bicyclists. V. A. Tucker,
American Scientist,
Vol. 63, No. 4 (July-August 1975), pp. 413-419

*Of course, most of the article is where I put "[...]". It's quite a good, fun read. There's even some kind of Galilean experiment with dropping pigeons and rats from heights.
A: To expand on Nicks answer, when you run, you sort of jump a little bit so that you raise your center of mass, which costs you energy equal to
$$\Delta E = m g \Delta h$$
Now when you lower your center of mass, the energy is dissipated as the vertical acceleration gained by going down is not increasing your horizontal speed.
This is definitely one of the causes.
Also, one can think about dissipated power, which might give us more insight. For example we have the following identity:
$$ P = \cfrac{\operatorname{d} W}{\operatorname{d} t} = \vec{F} \cdot \cfrac{\operatorname{d} \vec{x}}{\operatorname{d} t} = \vec{F} \cdot \vec{v}$$
Also, let's assume, that power losses in both cases are similar and that the input power is the same in both cases (we use similar muscles in both cases after all).
Now we have the velocity very different in both cases, and we could probably also agree, that there is much more force produced if we run as we can speed up to our maximum speed very quickly. So far everything agrees with the formulae.
Now we can easily convince ourselves, that even if the efficiency is similar in both cases, the loss of energy in running would be greater because we do it for longer for the same distance travelled. 
A: It's down to the biological inefficiency of maintaining the mechanical constraint of one foot stationary with regard to the floor, with the rest of the body moving. 
Let's suppose a skater on ice and another on rollers use approximately the same amount of energy to get to say 5 km/s. Both possess a different, but efficient compared to walking, mechanism that maintains that velocity, while satisfying any mechanical constraints. For the one on rollers, the point of contact between the ground and a roller must be stationary. For walking, the foot in contact with the ground must be stationary, and the biological mechanism for this introduces far greater biological losses.
However, there are far more efficient modes of biological transport such as hopping as in the case of Kangaroos, which uses half the energy of a marathon runner. Kinetic energy is converted into potential energy in the tendons while the mechanical constraint is maintained, with most converted back into kinetic energy upon leaving the gound.
A: *

*Bicycles make better use of inertia/momentum. As Nathaniel said, one push and you can coast for quite a while. That's just not possible while running.

*Running wastes energy moving up and down. In addition to moving forward, running requires a substantial upward push to get your body airborne, giving you time to bring your other foot forward. You then cushion and spring forward and upward again. While bicycling does have an up-and-down component to the pedaling, because the bike doesn't leave the ground, the energy you use in pedaling is converted much more efficiently to forward motion.

*Bicycling can translate weight to propulsion. While most serious bicyclists will tell you that pedaling is about spinning, not stepping, any 10-year-old can tell you that going butt-in-the-air and transferring your weight from left to right gets you going pretty quickly.

*Pedaling with toe clips makes use of the entire leg motion. When your feet are locked to the pedals, it isn't just the pushing-down portion of the pedal stroke that is used; lifting your foot, pulling it forward, pushing down and pushing backward all keep tension on that chain and so add power to the stroke. When running, fully half of your foot's cycle is wasted energy from a forward-motion perspective.

*Bicycling gives you mechanical advantage. Even with a single-gear bike, the motion of your foot is magnified when translated to the wheel. On a multi-speed bike, the ratio of the top gear is pretty high indeed. This allows two things; first, your effort is magnified, and second, your tempo slows, which reduces the amount of energy wasted moving the weight of your legs around.
A: A quick guess would be that for running you are lifting your centre of mass with every step while with cycling your centre of mass has a constant height.  Therefore you are only doing work against air resistance/friction in the bike mechanics.  While running you are also working against gravity.
A: The difference is in the underlying mechanism which transforms the chemical energy into kinetic energy of the vehicle or body. The answer is that the bicycle mechanism (due to having wheels etc..) is able to transform this energy better.
A classic analog is the lever or a pulley. One can use a lever or a pulley to lift a weight which would be very difficult (or even impossible) to lift with bare hands.
So to maintain same average speed with a bicycle one needs to use less chemical energy (than running) and as a result produce less heat.

"Give me a place to stand and I can move the earth"   

                                            -- Archimedes on the principe of the lever (allegedly)
A related answer on the principle of the mechanical lever (and variations)
A: As there are already answers which explained it very well, we can have an analogy to understand it further. 

Consider that square wheel is us and round wheel is cycle. Now let's try to rotate the square wheel. When you do, a normal force will act from the right edge of the wheel instead of center to prevent it from toppling. (Torque will act in opposite direction and stop it from rotating). So the only possible option is to pick this wheel in air and put on next step. 
But in case of round wheel, normal reaction always act at the bottom most point of the wheel and torque because of the normal force will always be zero. So there is no force acting on the round wheel to stop it's rotation. Because of this once a round wheel is given a push, will keep on rotating without any further force. (As there is no opposite force acting). 
Now if we compare both wheels, picking up a wheel and putting down on each step will always be more energy consuming or will require more work. 
A: I think the main reasons are:

*

*We have heavy legs designed more for climbing or slow walking where they can swing like a pendulum.

*We have to start & stop our legs using our muscles, whereas the crank on a bicycle recycles the momentum like the crankshaft of an engine recycles that of the pistons.

*Our muscles do not recycle energy fed back into them. In other words, we don't have regenerative braking. We actually waste a bit of energy bringing something to a stop, pushing it, or holding it up without moving, a bit like the resistance losses (copper losses) in an solenoid actuator.
While muscles are efficient, we don't use them efficiently when running.

