8-bit refers to the number of bits used in the data bus of a computer. It is the number of bits of data transferred on each read or write of the memory, and the number of bits used internally in the CPU to carry out processing. Similarly, a 4-bit CPU would process 4 bits at a time, a 16-bit CPU would process 16 bits at a time etc. A data unit of 8 bits is commonly called a byte (although the precise name is an octet), whereas a unit of 4 bits (a hex digit) is sometimes referred to as a "nibble".
8-bit processors normally use a 16-bit address bus, which means that their address space is limited to 64 Kilobytes; this is not a "natural law", however, and thus there are exceptions.