Expansion slots are used to add cards with additional and/or improved functions to your computer. In past decades, you needed to insert a modem card into an expansion slot to allow for Internet connectivity through the telephone cable. Also, you can insert a more advanced video card into an expansion slot for improved video performance.
Expansion slots come in many types and sizes, so it's important to verify which ones your motherboard has before buying any expansion card for your computer, and ensure compatibility. The most common nowadays are PCI and PCI Express.