PCI-Express
PCI-Express is the newest kind of expansion slot used in PCs. It is technologically superior to the older slots in every way. PCI-Express can be referred to using various names: PCI-Express, PCIe, or PCI-E. They all mean exactly the same thing. There's another completely different and incompatible bus called PCI-X so be sure not to get them confused. Despite the similarity in names, there's no hardware compatibility of any kind between PCI and PCI-Express. You can't plug PCI cards into PCI-Express slots or vice versa. It was just the computer industry doing their level best to confuse people. (Just for the record, the USB 2.0, USB High Speed, USB Full Speed naming debacle is the current leader in "the most confusing naming convention in existance" competition. The PCI-Express, PCI confusion is somewhat farther down the list.)
In PCI-Express x16, the "x16" part is pronounced, "times sixteen" or "by sixteen". The number following the "x" is the number of PCI-Express lanes in the slot. The more lanes in the slot, the faster it can go. The motherboard picture above shows both a x16 slot and a x1 slot. Video cards are normally designed to fit in x16 slots since they are the fastest. You can also get video cards designed for x1 slots. Those are normally used only if you want more than one video card in the computer. Most motherboards have one PCI-Express x16 slot for a video card and one or more x1 slots for other things like network adapters. Less common are x4 and x8 slots. You can "up-plug" PCI-Express cards. That means that you can plug a PCI-Express x1 expansion card into a PCI-Express x1, x4, x8, or x16 expansion slot and it will work (as long as the motherboard BIOS doesn't have bugs). The x1 expansion card can only run at x1 speed in any of those slots but it will work. Likewise, you can plug x4 expansion cards into x4, x8, and x16 slots and you can plug x8 expansion cards (if you can find one) into x8 and x16 slots. But you can't "down-plug" PCI-Express cards because an expansion card with a higher number of lanes (the "x" value) physically won't fit into an expansion slot with a lower number of lanes. For example, a x16 expansion card won't fit into a x8, x4, or x1 slot.