What @DocAwe said plus this TL;DR:
Serial, as in RS-232x, is a bidirectional serial data flow that uses a line each for TX and RX, and has a whole host of control lines - most of which you don't need if you're just doing simple transfers. It's unpowered, and relies on both devices being powered and understanding the protocol, as RS-232 relies on the devices to provide a method on the serial protocol.
PS/2 is a serial protocol, but it's specifically synchronous bi-directional powered. i.e. one clock line, one data line, one power line, and one ground line. You had different PS/2 ports on a computer because each port had to understand it's specific protocol (kbd, mouse, etc.)
What @DocAwe said plus this TL;DR:
Serial, as in RS-232x, is a bidirectional serial data flow that uses a line each for TX and RX, and has a whole host of control lines - most of which you don't need if you're just doing simple transfers. It's unpowered, and relies on both devices being powered and understanding the protocol, as RS-232 relies on the devices to provide a method on the serial protocol.
PS/2 is a serial protocol, but it's specifically synchronous bi-directional powered. i.e. one clock line, one data line, one power line, and one ground line. You had different PS/2 ports on a computer because each port had to understand it's specific protocol (kbd, mouse, etc.)