They are simply different interface types for graphics cards. Compatibility between any given video card and a motherboard is determined by the type of graphics interface they have.
No, you cannot put an AGP graphics card into a PCI-Express slot.
A PCI-Express interface only works with a PCI-Express graphics card. An AGP interface only works with an AGP graphics card.
The main difference is speed. PCI-Express is newer, and the highest PCI-Express speed is x16. The highest AGP speed is x8. Allegedly, PCI-Express is faster than AGP. However, it has been argued that there is not much difference in performance between these two slots.