x86-64 is the general name of a series of 64-bit processors and their associated instruction set architecture. These processors are compatible with legacy x86 software.
AMD and Intel are the major manufacturers, and VIA also supported it with VIA Nano processors. They call their own 64-bit processors AMD64, Intel64 and VIA x86-64 processors, respectively. Game consoles such as Microsoft Xbox One[1] and Sony PlayStation 4[2] use their own semi-customized AMD64-based APU as main processors.
x64 is another name used to refer to 64-bit extended systems[3] based on this kind of processors. People often use x64 to refer to its associated instruction set architecture and processor.