AGP and PCI are totally separate slots in any style of computer. Before AGP, video cards were made to go into PCI slots, though.
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A few questions, though: I have a 6600GT AGP card, which means I'd have to replace it. The system guide recommends the 'Sapphire Radeon HD 3850' - Is this a good replacement?"
Personally, I'd shy away from any ATI video card.
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Second, what's the difference between the various PCI slots?"
Night and day.
PCI-E is the new standard slot type, with lots of scalability built into the spec. Its meant to completely replace both PCI and AGP. Cards that need a lot of bandwidth (video cards, physics co-processors) will use the full 16 lanes of a PCI-E x16 slot, and low bandwidth cards (pretty much everything you'd use PCI for in the past) can use the lower profile, more efficient PCI-E x1 slots. The best part is that you can stick a PCI-E x1 card into a PCI-E x16 slot, and it will work perfectly. They also designed the spec to allow for bus bandwidth increases, and still maintain full backwards and forwards compatibility between devices and motherboards. Hopefully the industry will never have to change slots again (at least, until real computers start taking up 1 inch cubes

).
Most motherboards still sport a few PCI slots for backwards-compatibility purposes.
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Fourth, how would I tell if my old power supply would be enough for the new stuff?"
Plug it in, put the system under load, and see if its stable. If the components draw too much juice, the motherboard will shut off.
You could also try to figure out the power draw of all of the components and compare it to what the PSU is rated to put out. Just keep in mind that you should never run a PSU near or at max load, since the power output isn't very clean at that point (PSUs aren't really designed to deliver max load on a constant basis). How much power can your current PSU supply?
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Another thing to keep in mind is that you might not be able to use your current RAM, depending on how old your PC is. Unless you have DDR2 RAM already, you will probably need to pick some up. Luckily, it's dirt cheap right now, with rebates offered out the wazoo (at least in the US, not sure about Canadian market).