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eBay does categorize types although I will attempt to categorize again based on your need; starting with major categories AT/ATX power supplies (PSes).
AT: an obsolete traditional form (the actual variant is Baby AT the one that was mostly used) you have no choice in the matter if you are in need of an AT-style power supply, in fact there is no difference at all.
Specialized AT: just physically different, perhaps it was shaped in the form to fit in a Compaq chassis, if your computer is not a local unbranded build you have no choice in the matter either, if you are in need of such.
ATX: another major category, it superceded AT and has a completely different physical layout from the case down to where the CPU chip is, a regular ATX PS has a big motherboard power plug and a bundle of peripheral devices' power plugs, you may or may not encounter a small square plug for CPU fan.
Hybrid AT/ATX: just appeared on the market for a while when the industry was phasing to ATX engineering. Such a PS have AT-style plugs as well as ATX-style plugs. There should be no need to search for it, it was there in the first place so customers have some peace of mind knowing their chassis is gonna be up to date when upgrading to an ATX world.
No-switch ATX: there supposedly be a master power switch physically at near the power plug that look like -|0 if you don't, it still is an ATX through and through, but it simply meant electricity from the wall is coming in as long as it is hooked in. In replacing such ATX PS a regular ATX of any variant can do.
Athlon XP and Pentium 4 (and above) ATX: there is an extra plug in addition to the regular type. That is a CPU chip power plug, the first time I encountered such PS I dismissed it as a chassis fan plug; but if you make that mistake the Athlon XP or P4 will not boot up at all. It is a required plug for such class of machine. Such a PS is backward compatible and can replace a regular one, that plug becomes extra if not needed.
Active (has fan) PCI Express video card ATX: another extra plug for hooking up to the PCI Exp video card and powers its fan, required. Also backward compatible.
AT: an obsolete traditional form (the actual variant is Baby AT the one that was mostly used) you have no choice in the matter if you are in need of an AT-style power supply, in fact there is no difference at all.
Specialized AT: just physically different, perhaps it was shaped in the form to fit in a Compaq chassis, if your computer is not a local unbranded build you have no choice in the matter either, if you are in need of such.
ATX: another major category, it superceded AT and has a completely different physical layout from the case down to where the CPU chip is, a regular ATX PS has a big motherboard power plug and a bundle of peripheral devices' power plugs, you may or may not encounter a small square plug for CPU fan.
Hybrid AT/ATX: just appeared on the market for a while when the industry was phasing to ATX engineering. Such a PS have AT-style plugs as well as ATX-style plugs. There should be no need to search for it, it was there in the first place so customers have some peace of mind knowing their chassis is gonna be up to date when upgrading to an ATX world.
No-switch ATX: there supposedly be a master power switch physically at near the power plug that look like -|0 if you don't, it still is an ATX through and through, but it simply meant electricity from the wall is coming in as long as it is hooked in. In replacing such ATX PS a regular ATX of any variant can do.
Athlon XP and Pentium 4 (and above) ATX: there is an extra plug in addition to the regular type. That is a CPU chip power plug, the first time I encountered such PS I dismissed it as a chassis fan plug; but if you make that mistake the Athlon XP or P4 will not boot up at all. It is a required plug for such class of machine. Such a PS is backward compatible and can replace a regular one, that plug becomes extra if not needed.
Active (has fan) PCI Express video card ATX: another extra plug for hooking up to the PCI Exp video card and powers its fan, required. Also backward compatible.
Guide created: 10/11/06 (updated 12/09/07)
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