PCI came first, and you can still get some graphics cards for it, though they're rather weak.
AGP came next, and was a video interface...it's gone now.
PCI express, or PCI-e for short, is the new standard. The number after it, like x16, x4, or x1, refers to how many lanes of data the slot has. You'll notice that this corresponds directly to how long the slot it.
An interesting thing about PCI-e is that none of the connections behind the little tab on the front are dedicated to anything. What this means is that you can plug a PCI-e card into any PCI-e slot that it will fit in, no matter the size. Some motherboard manufacturers even leave the back of the slot open, so that you can plug cards made for bigger slots into the smaller ones. It's strange, but it works. Sometimes video cards even work there.
An exception to that is video cards and the PCIe x16
2.0 slots. Often, these are for use only with video cards, though it's not unheard of for them to be available for other cards to use.
Here's some pictures to help:
Here you can see the open-backed x4 and x1 slots that allow cards meant for larger slots to work: