What is a general definition of impedance? Impedance is a concept that shows up in any area of physics concerning waves.  In transmission lines, impedance is the ratio of voltage to current.  In optics, index of refraction plays a role similar to impedance.  Mechanical impedance is the ratio of force to velocity.
What is a general definition of impedance? 
What are some examples of "impedance matching" other than in electrical transmission lines?
 A: Uh... I liked this question. And a quick look at the Google or Google scholar did not give me much. So is written here more what I grasp people understand about impedance. Like Noldorin was saying I take impedance as how much is impeding power to be transferred from one place to another. In a homogeneous medium power is transferred as wave unimpeded but when it find an interface of another medium then power transfer won't be so straight forward.
The point is at the two mediums dynamic equations of the two place does no match. Say, electromagnetically one medium has the Maxwell equations of $(\mu_1,\epsilon_1)$ and the other medium has the equations with $(\mu_2,\epsilon_2)$
At the interface the countour conditions needs to be satisfied
\begin{equation}\left(\vec{D_2}-\vec{D_1}\right)\cdot\hat{n}=0\end{equation}
\begin{equation}\left(\vec{B_2}-\vec{B_1}\right)\cdot\hat{n}=0\end{equation}
\begin{equation}\left(\vec{E_2}-\vec{E_1}\right)\times\hat{n}=0\end{equation}
\begin{equation}\left(\vec{H_2}-\vec{H_1}\right)\times\hat{n}=0\end{equation}
where $\hat{n}$ is the normal of the interface. Using the above conditions to connect the solutions of wave equations at each medium, will give rise to the Reflections and Transmission Coefficients. In the case of Maxwell's equations
\begin{equation}R=\frac{Z_2-Z_1}{Z_2+Z_1}\end{equation}
For a one dimensional wave, with standard solutions
\begin{equation}f_i\left(x,t\right)=A e^{k_i x-\omega t}+B e^{k_i x+\omega t}\end{equation}
the conditions that needs to be satisfied are the equality of displacement and the its derivative
\begin{equation}f_2(x_0,t)=f_1(x_0,t)\end{equation}
\begin{equation}f'_2(x_0,t)=f'_1(x_0,t)\end{equation}
where $x_0$ is the position of the interface. Using these conditions with the solutions of the wave equations we will get.
\begin{equation}R=\frac{k_2-k_1}{k_2+k_1}\end{equation}
So in this case the impedance will be characterized by $k$.
So generalizing what you want is find what is the physical conditions that needs to be matched at the interface and use it to connect the solutions of each side of the interface, and figure how you get the reflections coefficients, which should look something like the reflection coefficients above. The terms should be your impedance for that system. Note that the definition of your impedance in that system is not quite unique. As in the previous example you could normalize the $k_i$ by some arbitrary $k_0$ and call this the impedance.
\begin{equation}z_i=\frac{k_i}{k_0}\end{equation}
Check the literature to find what is the exact definition for the system you are interested in. I suppose you already went to Wikipedia for other examples.
A: A good example of impedance matching outside the world of transmission lines is acoustic horns, which match acoustic impedance between a sound source (like a vibrating string or reed) and the air.
A: A good (if relatively grisly) example of impedance is the transmission of the shock-wave from an exploding bomb into a target. If a bomb detonates over a target, it will create a shock wave in the surrounding air, which will then hit and severely damage any structures it encounters. However, if the target is some sort of reinforced bunker, a large part of the energy will proceed to bounce off it as a reflected shock wave, because the acoustic impedance difference is too great.
This led to the development of earthquake bombs, which are designed to insert themselves into the ground before exploding; this causes the shock wave to travel through the ground and minimizes the impedance difference. A related "device" is the bouncing bomb of Dam Busters fame:

A: Mechanical impedance matching does have an application in electrical transmission lines (or any elastic cable/structure vibration) because it helps describe how much of the wave gets through a discontinuity, and how much is being reflected. Mechanical impedance is force over velocity and along the cable it is equal to tension over wave speed. The discontinuity may be either an elastic support (with damping properties), or a change unit weight of the cable (bigger cross section) which changes the wave speed.
For a cable with tension $T$ and unit weight $w$ (weight per length) the wave speed is $c=\sqrt{T/w}$ and the impedance is $z=\frac{T}{c}=\sqrt{T\,w}$. The practical application of this is in the design of stockbridge dampers that go on cables and structures to absorb aeolian vibrations. There is a theory of matching impedances and measuring the "Inverse Standing Wave Ratio" in testing to check the efficiency of the damper. With a good match most of the wave energy does not get reflected back into the cable once it reaches the end near the tower. There is a IEEE standard covering all this.
In case you missed it, mechanical impedance is force over speed. Maybe someone something similar for the electrical, or acoustical impedance.
A: I found a general, qualitative answer in David Blackstock's book Physical Acoustics, on page 46:

Impedance is often described as the ratio of a "push" variable $q_p$ (such as voltage or pressure) to a corresponding "flow" variable $q_f$ (such as current or particle velocity).  

I also received a nice answer to this question on another Q&A site which expands a bit on this qualitative statement with a quantitative one.  In particular, this answer makes the point that impedance is the ratio (transfer function) of a force applied at a particular point to the velocity at that point.  
I suppose what I am looking for next is an intuitive explanation of the phenomenon of impedance matching and maximum power transfer.
A: Impedance is general simply refers to the "amount of impeding (or block)". It's specific meanings in the various branches of physics (mechanical, electrical, wave impedance) varies, but are all loosely based on the concept of the everyday word.
A good example of impedance matching I always think of relates to audio systems. When you want to chain a number of amplifiers in stages, you usually want to match the output impedance of one with the input impedance of the other in order to maximise power transfer.
A: The key to the original question is the askers mention that it was always met in wave behavior situations-periodic functional solutions. The periodic behavior is a strong constraint place upon two paired variable which describe the system behavior. Some examples are voltage-current, $E-B$ component of electromagnetic fields, elastomechanical motion force-velocity. Generaly the product of the pair yields power (energy/time), but are constrained such that the ratio of the pair is at-most a complex constant-(the impedance or its reciprocal). Periodic functions have this property.
One responder suggest Force-velocity as a pair, but without qualification. Simple counter example is provided by a constant force. $Z = \frac{F}{v}  \sim \frac{1}{t^{2}}$. One might argue that it is true at least at $t = 0$, but at initial condition $f$,$v$ are uncoupled and independent. The utility of impedance lies in its constancy of proportionality.
The arguments the utility of impedance in power transfer problems misconstrued several major points by all sides. One often overlooked property of $Z$ is that it is periodicity dependent (i.e. on the wave frequency). Maximum power transfer occurs for source-load impedance match only over a limited bandwidth because generally the source and the load impedances vary differently with frequency. Match device must be retuned for large frequency change or else the matching range has to be increased with complex structures. Fine speakers and radio transmission line often get quite complex.
The arguments above also conflated harmonic balancing with maximum-transfer problem. They are intertwined but not identical. In a high fidelity systems, signal distortion almost always trumps the power efficiency issues. Thus eliminating distortion by reshaping the energy balance over the required transmission waveband is regularly practiced by shaping the $Z \left( f \right)$ distribution - i.e. by filtering.
Impedance is a useful concept when it expresses a constant (in time but not necessarily with wave-frequency) relationship between a pair of variables that characterize the behavior of a physical system. The product of these variables is the power transferred by the wave (or of their integral or derivative). Periodic functions satisfy the constant ratio constraint - Most non-periodic functions do not.
A: Riding a bicycle with multiple gears is an exercise in impedance matching. The right gear will maximize the transfer of the rider's efforts to the motion of the bike. Choosing a gear that is too high or too low (that doesn't match the output impedance of the rider to the input impedance of the bike) results in the bike moving more slowly.
A common mechanical example of impedance matching is waves on a string. Tie one end of a string to something solid and pull the other end taught. Now give the end you're holding a flick. A single wave will travel down the string. There will be a reflection of the wave when it hits the firmly attached end because that end has a higher impedance than the string itself. Likewise, if the other end of the string is tied to nothing like a bull whip (a very low impedance) then there will be a reflection too. OTOH, fasten the string to another piece with the same construction (and no big knot between them) there will be no reflection. The impedance is matched and energy is transferred efficiently.
Attach two strings of different size and density and there will be partial energy transfer and partial reflection.
An acoustic guitar will be loudest when the impedance of the strings matches the impedance of the vibrating part of the guitar AND there is a match in impedance between the vibrating guitar and the surrounding air. The guitar body is essentially an impedance converter between the strings and the air, much like the gears on a bicycle are impedance converter between the motion of the rider's feet and the motion of the bike down the road.
A: Impedance is the ratio of a force applied at a particular point to the velocity at that point.
