Audio gear usually has inputs and outputs.
Occasionally, the term "impedance" is used to describe an input or an output.
What does this mean?
INPUTS AND OUTPUTS HAVE DIFFERENT JOBS
Your inputs should a high impedance. That's a good thing. But like a lot of good things, too much can be bad. Really high impedance inputs can pickup noise and even radio signals, so designers will lower the impedance when they can. As a rule of thumb though, we want our input impedances to be high.
When you have just enough impedance for an input, it should pass the audio without attenuating, coloring, or distorting the audio in any noticeable way, and without adding noticeable noise.
Your outputs should have a low impedance. This isn't a problem with modern gear. Sometimes a piece of vintage gear or a copy of a vintage circuit will have a high output impedance. This is undesirable.
When an output is said to be "low impedance" it implies that it has power and can drive many signal paths, or perhaps even a speaker.
This "high input" and "low output" system is known as impedance bridging and it has been standard since transistors began taking over electronics, many decades ago.
IMPEDANCE MATCHING
It is often repeated that you can get "maximum power transfer with matched impedances," or something along those lines. This is mostly of historical interest, but does apply to those who use tube power amplifiers with output transformers.
In modern signal processing, we couple circuits together so that only voltages transfer; we don't care about the current usually, and we certainly don't care about the power. We want our voltage to be accurate, and we want our current and power to be as low as we can practically make them. Getting maximum power transfer usually attenuates the voltage, and this is undesirable in modern systems (barring "creative" audio engineering practices).
There was a time when audio signal impedances were matched. Before the transistor revolutionized electronics in the 1950s and 1960s, vacuum tubes and transformers were the kings of the audio path. Transformers were used as the standard input and output device. A 600Ω microphone would connect to a 600Ω microphone input transformer. A 16Ω loudspeaker would be driven by the 16Ω tap on an amp's output transformer. While microphones are usually not matched anymore, output transformers and loudspeakers still are.
Signal impedances need to be matched when using very, very long cables. Once the cable length is longer than 10% of the wavelength, matched impedances become a real requirement. For audio, this length is about 3500ft. Are you running a cable that long? Thanks to cheap solid state amplifiers, we no longer worry about matching impedances in modern audio installations.
For your cable internet connection, there is a MHz carrier wave traveling along that coax wire that comes into the house and snakes over to your modem. A MHz frequencies, the wavelengths are so small that even a few feet require impedance matching. Don't mess with your cable wiring if you don't know what you're doing. An impedance mismatch will cause reflections that will slow down your internet service. Digital cable runs do have to worry about impedance and matching. To avoid this, most digital cables (USB for instance), are only sold in fixed, small lengths. There is a reason for this, and that is to avoid reflections and attenuation that could cause data corruption. To make a long USB cable run, you have to use a USB hub that will effectively copy the information and send it down the next length of wire.
IMPEDANCE MISMATCHING
What happens when we reverse our "bridging" system and plug a "high output" source into a "low input" load? We can expect some attenuation and/or distortion from this. Some combinations will favor high end attenuation first, and other systems may favor low end attenuation first. In design and engineering, this type of impedance mismatch is considered to just be bad. In the more creative world of audio recording, such frequency dependent attenuation and distortion might actually be desirable to certain producers, so many mic preamplifiers will have a selectable input impedance control to purposefully mismatch.
When it comes to output transformers, an impedance mismatch of the "bridging" type (8Ω output transformer tap trying to drive a 16Ω or greater speaker load) may cause serious damage to the transformer. Raising the speaker impedance and driving the same input signal will increase the voltages observed on the output transformer's secondaries. If these voltages exceed the design limitations for too long a time, permanent damage can result. The two rules of thumb with power amps and output transformers are 1) open circuits are worse than short circuits, and 2) it is always best to simply match the power amp's stated output impedance to a matching speaker cabinet.
Transformerless power amplifiers (most modern amplifiers) work on the voltage drive principle, and thus work in "bridging" mode. They will have a very low output impedance (<1Ω) and will drive different loads so long as the load does not exceed a minimum load (typically 4Ω is the minimum, some amplifiers are rated for 2Ω minimum).