I have heard many vehicle manufacturers have been using this to improve power and torque figures.
They do this by altering the length of the manifold, I am hoping for better diagrammatic explanation.
The length of the intake runners have certain affects on the engine operation. For example, longer intake runners are used to improve the bottom end torque (torque at low RPMs) while shorter intake runners will improve top end power (horsepower at high RPMs). The lengths will vary from engine to engine as well as what the goals of each vehicle the engine will be powering.
You also have to take into account the diameter of each runner. Everything correlates to mass and velocity of airflow. Through the RPM range the velocity of airflow in the runners increases, but at some point, it maxes out and cannot go any faster which will be limiting. As the velocity of the airflow increases, so does it's inertia. At the bottom of the intake stroke, the inertia of the airflow will help push a little more air into the cylinder which will help power. But, if the runner isn't optimal, that can't happen.
For example, a long, smaller diameter runner will help low end torque because it will hit the velocity limit earlier, but it will hurt top end horsepower because it's too restrictive. A short, large diameter runner will help top end power because it will reach maximum velocity later, but it will not help low end torque because it cannot get enough velocity to build inertia.
Now that you know what the differences between the runner lengths are, you can imagine why having a variable length runner manifold would be a good idea. You get the best of both worlds. With a normal manifold, you have to pick the exact manifold for whatever your goals are. If you plan to do a lot of drag racing, then probably a manifold with runners to support top-end power would be best where as if you're doing autocross where you're down in the low-end RPM range most of the time, you might choose a manifold to best improve low end torque. It all varies and there is no right or wrong answer for all applications. But there is going to be a compromise.
For the regular daily driver though, you don't want to compromise because you need the low end torque for driving around stop light to stop light, but also you need that power on top for merging on the freeway or passing someone.
The variable length runner manifolds use a valve to switch back and forth between two runners depending on what the situation calls for. When the engine load is high (low RPM), the manifold will switch to use a longer, smaller runner to get the low end torque. When the engine load is low (high RPM), the manifold will switch to use a shorter, bigger runner to help with power on top. The best of both worlds.
Disclaimer: This is a simplified explanation of intake manifolds and runners. There is a whole world of science out there between surge tanks, more airflow dynamics like turbulence, swirl, etc and of course when it comes to forced induction engines (turbo, super chargers), these rules change.
Edit: Here is an image
You can see as described in one of the comments, there is a shaft that controls a set of butterflys. The shaft will rotate which will re-position the butterflys effectively changing the runner properties. The shaft is vacuum modulated in this picture as you can see (start the the linkage and work to the left). There is a bell-shaped modulator with a vacuum line attached to it. Modern ones may use more electronic methods.
Variable-length intakes increase the pressure of the air entering the intake manifold thanks to a physical phenomenon called Helmholtz resonance.
It's also known as dynamic supercharging since it avoids the use of a mechanical device (compressor/blower) to boost intake air pressure.
Without getting too technical, any air intake geometry has a certain Helmholtz frequency associated with it, just like how blowing over the neck of an open bottle produces a certain note or pitch.
At this frequency, the air molecules vibrate more, resulting in higher pressure.
Engine RPM will govern how often the intake valves open and shut. These valves generate pulses that translate to a frequency signature.
The idea behind varying the effective geometry is to get the Helmholtz frequency of the air intake to sync up with the frequency demanded by the engine over a range of RPMs.
This causes intake air to enter the cylinders at a higher pressure. Needless to say:
▲ Air Pressure → ▲ Bang → ▲ Torque → ▲ Power
There are many ways, each with their own advantages and disadvantages:
Lengthening/shortening the intake runners
The '91 Le Mans-winning Mazda 787B is an early example of this; the linked YouTube video shows the intake runners sliding up and down like a trombone.
▲ RPM → ▼ Length required
Regulating between two intake runners of different lengths
This is what DustinDavis' answer describes. Imagine air flowing through two intake runners, one long and one short.
At the end of the runner, a butterfly valve determines how much air is drawn in from each runner in turn. Changing the position of the valve changes the effective intake length
Oscillatory intake systems
These setups use the opening and closing of intake valves to control the effective geometry of the intake.
Often the costs outweigh the benefits. As much as we may wish it, power isn't everything.
Plus, this setup offers only modest power/torque gains. Typical gains will be in the 3-5 % ballpark with this approach.