I'm sure that somebody can answer these longstanding questions for me.
I can't tell if you're looking for a technical explanation, or just wondering what to look for in selecting an AVR/amp.
Lots of technical discussion already provided.
If you're asking what it really means to you when shopping for an AVR/amp, the answer is more simple. You may have gathered there are a lot of variables and different ways to express power. Unless you are technically savvy, you really can't decipher the truth from their published numbers.
However, you can look for the most common way to report power, and use it to roughly compare products. I think most products will give you an X watts - 20-20k Hz - 2 channels - 8 ohms rating. When comparing product specs, make sure all their qualifiers are the same. If you see different qualifiers, know that maybe a comparison is not valid.
Also realize that power is not the only, and maybe not even the main parameter to consider. Your room size, listening preference, speaker sensitivity, and more are all factors that can make more difference than power.