When you buy a computer, it comes from the manufacturer with the capacity to perform certain tasks out of the box (i.e, word-processing, playing games, etc.). However, certain specialized functions, such as professional quality audio and video manipulation, usually require that you add both software and hardware to the computer.
Hardware is usually "added" to the computer by means of expansion slots, which are connectors mounted on the motherboard that allow for the installation of cards that the computer can use for the specialized functions. The standard expansion slot format used in computers today is called PCI (which stands for Peripheral Computer Interface); Thus, a PCI slot is simply a computer expansion slot that complies to the PCI standard.