What is the 15-pin black port that old sound cards have?

This is a game port.

It was originally designed by IBM to support arcade-style analog joystick controllers, but was later extended by Creative Labs (the company behind Sound Blaster) to additionally support MIDI devices such as synthesizers through an adapter.

A single game port supports two joysticks, each with two analog axes, and a total of four digital buttons.

USB would later replace the game port for both of these applications. Needless to say, USB is now used for myriad other purposes, from keyboards and mice to storage devices and battery charging.

It's a game port (as Ramhound points out correctly). Used to connect a MIDI device (like a keyboard) or joysticks before USB was introduced.

Edit (added from comments): It was originally located on a dedicated expansion card and later integrated into sound cards (like the Sound Blaster card you are showing on your picture).