In order to place a single character on the screen (for example, the letter 'A'), a computer needs to process 8 binary digits - 8 bits of data. Thus, an 8-bit computer can deal with one character, whilst a 16 bit computer is powerful enough to deal with two characters simultaneously. Similarly, a 32-bit computer can cope with four characters and a 64 bit machine can process eight characters.
In the early 90's, 'bits' were often used to indicate the technical capabilities of a console. For example, the NES was 8-bit, the Sega Mega Drive was 16-bit, whilst the PlayStation a 32-bit console. However, Bit ratings for consoles largely fell by the wayside after the 32-bit era, as there was little to be gained from increasing the word size much beyond 32-bits; performance depended on other factors, such as processor speed, graphics processor speed, bandwidth and memory size.