BJL,
The problem of noise in general is a concern, whether we're referring to images, music, traffic, neighbours, meditation, and/or just getting a good night's sleep.
Noise is any unwanted and undesired stimuli, usually with an emphasis on sound, but in the case of photography, an emphasis on the visual.
I think the issue of noise in images has come to the forefront because digital processing and the computer have allowed us to examine in great detail all the technical deficiencies in our images.
With that marvelous tool, Photoshop, we can attempt to raise any dark shadow in any part of an image, to see what interesting detail might lurk there. If the detail we see is bloody banding (vertical or horizontal streaking of thin lines), or just colorful mushiness of an abstract nature, then we might become displeased. At least, I do.
I know such deficiencies might be irrelevant in a postcard-size print, and sometimes even on an A4 or A3 size print. However, for me, that's not the point. Technology is always advancing. In 10 years time, I might buy the latest 100", 8k, OLED TV screen. I'll then start reprocessing the best of my images for display at that resolution, and I don't want to see noisy crap in the shadows, or alternatively, totally black shadows devoid of detail.