Impedance is the resistance* to the flow of current. High power amplifiers are rated to be stable into some impedance. If have nothing connected to the output of an amp, that air between the terminals presents nearly infinite impedance. On the other hand a piece of wire running between the terminals would be next to 0 impedance.
An amp doesn't like to have it's terminals shorted, but the reason may not be immediately obvious. What actually happens when you connect a current source to ground is the flow rate gets so high the electronic parts making up the device can't handle that much, so they overheat and fail.
So amps are rated for the minimum impedance which they can support. At the other end, a higher impedance doesn't really hurt anything, it just decreases the over-all efficiency of the speaker, thus requiring more power to play at the same volume level as another similar speaker of lower impedance.
I don't follow headphone specs too much, but the same applies there, just the amps are designed to run into higher impedance drivers while still presenting high enough listening levels.
As for the input impedance of a device. With some equipment, the output when connecting to another device must have a "matched impedance". This doesn't apply for the output of an amp driving a speaker, where the output is in the kΩ range (although it may have some effect on the dampening factor). But when connecting the output of a pre-amp to a power-amp, those two should be matched. Without getting too technical, it's basically like they have to be speaking the same language to communicate effectively. It isn't something really to worry about in consumer gear as everything with RCA jacks will work together.
*Resistance is actually the measurement of voltage loss through a circuit.
