Any kind of electrical power transfer has a typical ratio of voltage to current. For example, you can deliver 100 watts by 1 amp at 100 volts, or 10 amps at 10 volts, or any other product that comes to 100. It's convenient to express the ratio of volts to amps as a number of ohms (since dimensionally that's all ohms are anyway). Power sources, loads, and even transmission lines all have characteristic impedance, which tells you something about what will happen when things are connected together.
Impedance matching is the selection of components with identical impedance, or the addition of impedance transforming components to cause a component with an undesired impedance to appear as though it has a more desirable one.
As Brian Carlton pointed out, when you have matched impedance, you achieve maximal power transfer. This is often desirable, but not always. The thing to remember is that maximal power transfer is achieved at the cost of dissipating equal power in the source and the load.
So for example, a case for not matching impedance is when you want to efficiently use the energy from a source. A 0.1 ohm load would get optimal power out of a battery with a 0.1 ohm internal resistance, but half the energy would be dissipated in the battery itself, which would be rather wasteful of the stored energy. (Not to mention that the terminal voltage would fall to half!) By purposely using a load much with much higher resistance than the battery, most of the stored energy ends up doing work in the load.
On the other hand, you DO want to match impedance when, for example, you have an audio amplifier stage that ideally wants to drive a 600 ohm load, but you only have a 3.2 ohm speaker. An ordinary transformer, having a 1:N voltage ratio, will conveniently give you a 1:N^2 impedance ratio. Another common case is in RF work, where as volting pointed out, you need to minimize reflections, because reflected energy can cause excessive power dissipation in your source.
Yes, you can build differential (balanced) matching network, it is a bit more complicated, but general rule is to divide series impedance by 2 (there are 2 series components, one on the positive side, one on the negative side) and multiply shunts by 2 compared to unbalanced Pi or T networks. mind you, component tolerances become very critical as mismatch between the positive and negative lines can cause unwanted spurious emissions, especially with PAs.
Edit:
A tech note from TI http://www.ti.com/lit/an/slwa053b/slwa053b.pdf
Best Answer
Unfortunately, you cannot match a purely reactive load. You could cancel out the reactance at a particular frequency using an inductor where $$Z_L=j*\omega*L$$ and $$ Z_C=\frac{1}{j*\omega*C}=\frac{-j}{\omega*C}$$
However, that would just give you a short circuit. In any case using the formula for reflection coefficient of a load on a 50 ohm line $$ \Gamma=\frac{Z_{Load}-50}{Z_{Load}+50}$$
The magnitude of Gamma for any purely imaginary Z_Load will be 1 (complete voltage wave reflection). For example, if Z_Load is j50, you'll get a Gamma of j. If Z_load is j25 you'll get a Gamma of -0.6 + j0.8. If it's -j33 you'll get -0.39-j0.92, etc.
I would just try to keep your transmission line very short and hope your source has sufficiently low output impedance to deal with this very non-ideal load.