Bits as a computing term started out as what number of bits registers in the CPU could access in general operations. In general computing and electronics (worlds not so very far apart at the time) it also counted for something as shifts were happening.
Marketing later twisted it almost to breaking when certain processors, buses, DMA and whatever else might have technically supported more but the general day to day operations would not have.
http://www.zdnet.com/article/ects-the-truth-about-the-dreamcast/ for an article from the time, and a less well researched one from about the same
http://edition.cnn.com/TECH/computing/9909/09/dreamcast/ and the Jaguar was already brought up for those looking for an early preview of that world.
Today it is mostly a shorthand for a graphical style, and maybe audio as well. It is short, pithy and dodges both regional naming issues and the amount of devices playing at the time. The differences between said styles were fairly visually stark to almost anybody though -- more pixels in the average sprite, more colours (certainly more colours in heavy use), things like parallax scrolling, on screen sprite counts in some cases, number of frames of animation in others.
When you are taught about contractions, apostrophes and similar things you are often taught to expand the sentence (its vs it's you will be told it's is short for either it is or it has so if your sentence does not make sense with one of those you don't want it), by similar token if a phrase like "it attempts to evoke a ? bit aesthetic/style for its blah" makes the sentence make more sense then you have it. It can also be used in a negative for many things have improved considerably in the time since. I have never been a fan of the generations nomenclature (way too many fuzzy edges, and that is before you consider the PC when there were notable differences beyond free online and mods) and prefer era, hence phrases like "the company was dominant during the 16 bit era".
For audio
It covers some good stuff there.