The PCI expansion slot uses a shared PARALLEL bus architecture, where the PCI Express uses point-to-point serial links.
To further understand this, let's look at the computer bus,
http://en.wikipedia.org/wiki/Bus_(computer)
Parallel communication,
http://en.wikipedia.org/wiki/Parallel_communications
and Serial communication,
http://en.wikipedia.org/wiki/Serial_communication
Here's info about PCI Express,
http://en.wikipedia.org/wiki/PCI_Express
Now you can go back in technology to the PCI bus,
http://en.wikipedia.org/wiki/PCI_Local_Bus
No, you cannot use a PCI Express graphics card in a PCI slot.
They are two different technologies, for one.
Two, the slots are different. The contact pins for the voltage and signal are in a different place, as well as the physical slot size difference, plus the locating lugs down in the bottom of the slots.
Edit: Also, no is there isn't an adapter to change a PCI slot to a PCI-Express x16 slot.