Why Are the PCI Express Ports on My Motherboard Different Sizes? x16, x8, x4, and x1 Explained

The PCI Express standard is one of the staples of modern computing, with a slot on more or less every desktop computer made in the last decade. But the nature of the connection is somewhat nebulous: on a new PC, you might see a half-dozen ports in three or four different sizes, all labelled “PCIE” or PCI-E.” So why the confusion, and which ones can you actually use?