Decoupling: An ideal decoupling element is a lossless path for direct current and an infinite impedance for alternating current ... this allows devices to be powered with no loss and prevents any noise from moving from one load to the next. 
Bypassing and Decoupling are subjects that are rarely, if ever, covered in the classroom; but are emphasized in nearly ALL Data Sheets & Application Notes! 

The legacy of this is that many new engineers and some technicians ignore it, but they do so at their peril!

It is the single greatest reason for hardware debugging problems!

Decoupling is a refinement of bypassing. Because of bypassing's finite limitations in creating the ideal voltage source," the decoupling, or isolation of adjacent NOISE sources is often required.
As with bypassing, the reason for decoupling is to prevent unwanted communications between different devices (or different stages of the same device) that share the same power rail.  Also, implicit in this requirement, is the reduction of NOISE present on that power rail--at all frequencies.
Where as in the case of bypassing a shunt capacitor was placed across the power rail, decoupling completes the implied "RC" (LC) part of the network: the series element--as in a lowpass filter. 
Ideal Voltage Source:
Furnishes a constant voltage output, regardless of the load or load variations; the output Impedance = zero ohms at all frequencies and all conditions.
Classic Decoupling using a "LC" Network, A.K.A. a Lowpass Filter.
Not-so Classic method of Decoupling, using a Voltage Regulator in place of the LC network.
The "ideal" decoupling device is a lossless path for DC
and an open circuit to AC.
Series Resonant  R L C  Circuit
(Notice the effect of increased R, red & blue plot)
NOISE Attenuation of Decoupling & Bypassing

Parallel Resonant  R L C  Circuit
(Notice the effect of increased R, red & blue plot)
Low Pass Filter = Decoupling & Bypassing
R L C   Filter Refresher 

There are instances where the power distribution between stages cannot be sufficiently bypassed. In this case, the designer might be tempted to use several different power supplies. However, by supplying the DC power to each stage through a separate inductor or "RF choke," while also bypassing that stage to ground, the effect can be nearly the same. That is to say, the choke offers a high impedance path to any errant signals or noise between stages, while offering a very low resistance path to the DC power: this is known as decoupling. Active devices such as voltage regulators can also be used for decoupling stages. 

In fact, considering the size of inductors as compared to surface-mount voltage regulators: regulators might be the better choice. One might better understand this by recognizing the fact that a choke or inductor is one of the two needed components for a Resonant circuit. Therefore, the combination of decoupling inductor and bypass capacitor could just happen to resonate at the wrong frequency. Having said that, it might be obvious that the inductor needs to be as small a value as is reasonable, and the bypass capacitor as large as practical. This is essentially correct, however, there is still the possibility that the resonant frequency of this combination could cause mischief. And, if that weren't enough, the inductor can be self resonant. This is caused by the distributed capacitance between windings, i.e., one turn of wire to the adjacent turn of wire, etc...

Powering both Analog and Digital
As in the figure, there are instances where both types of Decoupling are appropriate.
(Especially when powering both Analog and Digital from the same Power Supply)
RC Amplifier with various degrees of Power Supply Stabalization
in the form of Bypass & Decoupling
Sans Bypass Capacitor
With One Bypass Capacitor
With Two Bypass Capacitors
Two Bypass Capacitors & Decoupling Resistor
Note: in low current stages and devices, designers often replace the decoupling choke with a cheaper decoupling resistor to save money. 
---More Discussion:
Bypassing, decoupling, shielding and groundplanes are the elements that allow circuits -- analog and digital-- to function properly. The reason is simple: let's say you have a cascade of amplifier stages (see figure below) that are boosting or amplifying an otherwise weak signal. The input is very sensitive to small signals, and successive stages are drawing progressively more current in order to produce the larger replica of this weak input signal. In doing so, the output stage draws large amounts of current at varying rates. This large varying current is seen by the more sensitive input stages through the common power supply rail, which serves all stages. This can happen if the power rail, be it wire or PCB traces, is of sufficiently high impedance. Even if the power supply were "ideal," (zero ohms) this can still happen: as the frequencies go higher, the inductive reactance, of the leads or PCB traces, increases. For example, if some fast transitions of the input signal caused a resulting perturbation on the power supply rail to propagate down that rail to all of the other circuits, the resulting effect can be oscillations or some sort of instability which could cause distortion or even render the circuit inoperative. One can think of it as inappropriate feedback between stages, facilitated by the power rail not appearing as a Virtual AC Ground.
 Bypassed & Decoupled Distributed Power to Cascaded Amplifier Stages
(In essence is a "n" order Lowpass Filter)
RECAP:   Intrinsic Inductive Reactance (XL) in Power Distribution 
The Ideal Voltage Source furnishes a constant voltage regardless of the load, or load variations; the output impedance = zero ohms at all frequencies and all conditions.
Power Rails
Notice how the "Intrinsic" inductive reactance (XL) of the finite length conductors effects this zero ohms impedance.
Adding bypass capacitors to the intrinsic inductive reactance helps to restore the (virtual voltage source) zero ohms impedance.
Adding even more Inductance in the form of a "Decoupling Inductor" isolates the bypassed device(s), further helping to restore the (virtual voltage source) zero ohms impedance.
-- Even More Discussion
One more thing to consider about chokes: the "Q" or quality of the inductor has an effect on its efficiency. As previously stated, the inductor should appear as a perfect path to the DC power it is carrying, and a high impedance to any AC, i.e., no series "R." In the practical world this isn't feasible. However, if heavy current carrying chokes are required, then the choke must have higher "Q," i.e., less wire which means lower "R." This can be achieved by using chokes with ferrite cores, which need considerably less wire for the same value of inductance.  The core can be viewed as a multiplier of "Q." 
Small donut or tubular shaped ferrites called "beads" are regularly slipped over leads to act as decouplers.  One popular use for ferrite beads is to suppress parasitic high-frequency oscillations in circuits where the gain device (such as a transistor) has significant high-frequency gain. 

1) One of the most efficient inductors is the ferrite toroid. It has high "Q" -- low "R" -- and because of its toroidal shape its fields are confined which minimizes its interaction with other circuit elements. 

2) There is always some decoupling built into any circuit.  The power conductors act as decoupling inductors. Although short trace lengths are usually desirable, longer ones sometimes actually improve decoupling.  However, don't forget that long ground traces or conductors never help ... they make effective bypassing impossible! 

3) The down-side of ferrite, is that it will change inductance as the current or flux changes. In the case of large currents, it can saturate. However, by correct component choice -- frequency, AC and DC current, etc. -- ferrite is a great tool for the designer.

1) LC decoupling is used where the supply voltage cannot be lowered, i.e., if one needed a noise-free +12 volts on a PC bus, say. One could get a "clean" +12 volts with a voltage regulator... if only there was +15 volts or higher to start with. But such is not the case. So you use a high "Q" inductor (RFC choke) along with the proper bypass capacitors to effectively lowpass filter the +12 volt supply rail. 
   For a very noisy supply you can use more than one network, i.e., one or more "pi" networks.

2) One of the most efficient inductors is the ferrite toroid. It has high "Q" -- low "R" -- and because of its toroidal shape its fields are confined, and therefore has less stray fields. The super star of high "Q" inductors or transformers is the pot core. And of course, don't forget the ferrite bead. Thread the wire through the bead once or several passes and it may be just what the doctor ordered. 

3) Decoupling is only as good as the components that you use. The capacitor part of the network should be high "Q" and minimum inductance: the noise is dropped across the inductor, and the capacitor must exclude the remaining noise. Another way of saying it: in a perfect world the inductor is an open circuit to noise (AC) and the capacitor is a dead short -- Zero, Nada, Caput, Zilch; "This here parrot is dead." The slightest inductance in series with that capacitor, and some very high frequency noise will come through like Gang Busters! 
4) SMT or chip capacitors made of ceramic are best. Sometimes in critical circuits, several size caps in parallel are appropriate, e.g., 1ufd || .1ufd || .001ufd, etc. The reason for this is as the capacitors become smaller in value, they also get physically smaller, hence less inductance. However this is less the case with SMT caps: consult your capacitor data sheets for the impedance verses frequency plots.
See impedance verses frequency plots.

Linear Voltage Regulators as Decoupling Devices
Simple Shunt Regulator
Simple Pass Regulator
Pass Regulator with Gain
Linear Regulators 

The use of three terminal linear voltage regulators, like the 78xx and 79xx devices, is fairly straightforward. However, there are a few things to remember: always bypass -- there's that word again! -- the input pin to the common pin with a ceramic capacitor no smaller than 0.33 ufd, and use absolutely the shortest leads possible (there are some transistors with pretty high ft in that regulator, and if you furnish enough reactance of the wrong kind, Mr. Oscillation will visit).
Points to Ponder:

1) Unlike bypassing, decoupling is not always used; in fact, it is typically used as a last resort if bypassing fails to give the wanted power supply stabilization.

2) There is always some decoupling built into any circuit: the very inductive reactance bypassing is meant to overcome--the power conductors--in essence act as decoupling inductors.
  a_ Although short trace lengths are desirable, the power lead being long can, sometimes, actually improve decoupling. 

3) The simple addition of ferrite to these conductors enhance this natural decoupling inductance; sometimes eliminating the need for a special decoupling inductor.

4) The best all-round decoupling device is in fact not an inductor, but a voltage regulator--especially LDOs (low drop out regulators).

"Decoupling" Power Distribution Bus Bars