What does 8-bit / 16-bit actually refer to?
Why is NES is called an 8 bit console? Theres also a SNES which is called 16 bit console. What are these bits?
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
The NES comes from the "8-bit era" of consoles, which included the NES and the Sega Master System. These consoles are based around 8-bit processors, which generally store and process data 8 bits at a time.
In computer parlance, 8 bits make one byte. Many NES games have a limit of 255 on certain items (such as The Legend of Zelda's Rupee counter) because 255 is the maximum unsigned integer that can be stored in 8 bits of data. In the early days, this was a serious limitation on computer technology, and many workarounds were needed in order to program even moderately complex games like Punch Out! that included large sprites with many frames of animation.
As computer technology evolved, so did the processors available for video game consoles. The SNES featured a 16-bit chip, the PlayStation a 32-bit chip, and the Nintendo 64 featured a 64-bit chip.
However, the number of bits the processor generally handles at a time is only one measure of the performance of the system; the original Xbox was based around a 32-bit Pentium III chip 5 years after the release of the Nintendo 64. At the same processors were increasing in complexity, their overall computing power increased as well, and the cost of storage (both in terms of memory chips and optical media) declined.
The intersection of these technological improvements and the rise in popularity of video games as an entertainment media is what caused the increase in complexity and graphical quality that we see in video games today.
The "8-bit" moniker used to describe the NES and its peers is the same distinction we talk about today when we talk about a "64-bit" PC processor or "64-bit Windows" - these architectural terms are ones we use in more areas than just video games.