First a note on sloppy terminology: When talking about a computer system it is best to refer to it as a "system" or a "computer" -- never as a "CPU" (a CPU is a processor) and preferably not as a "box." Nearly everything in a computer could be described as a "box." To better communicate your intentions, refer to the enclosure that your system uses to house the internal components as a "chassis" or "case."
Power and size standards
AT and Baby-AT: The official design specification for the Baby-AT formfactor is no longer listed at The Desktop Formfactor site, but I can tell you for certain that the maximum size for an AT motherboard is 12"x9.6" (this is the same as ATX, but turned 90 degrees). The Baby-AT and AT formfactors were phased out during the Pentium-Pro era, and by the time the Pentium3 was introduced, the ATX standard had completely taken over for both motherboards and power supplies. Many motherboards, power supplies, and cases of that era have the capability of using both the AT and ATX standards, although a transitional power supply wouldn't conform to the current modern ATX specification. Aside from the power standard, the differences between AT and ATX power supplies are the type of connectors used (AT power supplies had two six-wire connectors that fit on the motherboard side-by-side) and the method of activating the power switch. AT power supplies use old-style click or rocker switches that mount on the front of the case; ATX power supplies are activated through a signal that is carried through the motherboard from a jumper switch on the front of the case.
AT refers to the power supply and Baby-AT refers to the motherboard formfactor. Both are obsolete.
ATX: Basically ATX is the same architecture as Baby-AT except it's turned 90 degrees, has a more advanced power supply connector, mounts the CPU away from the expansion slots to accommodate longer PCI cards, and has side-mounted I/O connectors hardwired onto the board. Basically this design was implemented to make computers cost less by having more integrated peripherals and to cut down on build times and electro-magnetic interference (EMI) by eliminating a lot of the internal cabling. The official ATX specification is 12"x9.6", but manufacturers have the option of making the board smaller while still retaining the ATX standard. For that reason you'll find that many ATX boards are not exactly 12x9.6.
ATX also refers to the power supply and chassis requirements that an ATX motherboard needs. An ATX power supply must deliver a certain amount of power at the correct frequency, and it must have the proper connectors. Modern Pentium4-approved ATX power supplies have a four-pin 12V connector that provides extra power for the voltage regulator. This is required on all Pentium4 boards and recommended for some of the newer AMD-based motherboards that have the extra 12V socket. Asus motherboards generally have a standard Molex (drive) power socket on the motherboard that allows you to give power to the voltage regulator using a regular drive power cable even if you don't have a power supply that has one of the newer-style 12V connectors. You cannot put an ATX motherboard into an AT case because of the drastically different board design, although there are some hybrid cases that can support either formfactor. All ATX derivatives are backwards compatible; a FlexATX, Mini-ATX or microATX motherboard can all easily fit into a full-size standard ATX case.
Mini-ATX: A derivative of the full-size ATX motherboard standard. The maximum size is 11.2"x8.2" although motherboards can be smaller than that and still fit the standard. Mini-ATX motherboards are not all that common, although there are still new motherboards out there that conform to this size standard. It is functionally no different than the ATX standard except for its slightly smaller size; Mini-ATX motherboards generally have very few onboard peripherals and can have a somewhat cramped board design which makes them difficult to install.
microATX: The microATX standard was introduced to allow highly upgradable motherboards to fit into smaller chassis designs with smaller power supplies. The official size standard is 9.6"x9.6" but many microATX motherboards end up being a little smaller than that. MicroATX motherboards generally have three PCI slots and one AGP slot and have many onboard peripherals and features. A microATX motherboard can easily fit into any ATX-compliant chassis. Other than these slight modifications in design, there is little functional difference between an ATX and microATX motherboard. There is also a microATX power supply standard, but they are not nearly as popular as standard ATX supplies. The only reason you'd want to use one of the smaller, less powerful microATX power supplies is if that's all that will fit in your case -- mini-desktop cases generally require microATX power supplies because of their small size.
FlexATX: The FlexATX standard is a derivative of the microATX standard. It is smaller, uses less power, and has greatly reduced add-on potential. Basically a FlexATX motherboard is designed to be an all-in-one solution that you install memory, a CPU, and a hard drive on and it's ready to go. The maximum size for a FlexATX motherboard is 9"x7.5" although, as previously stated, it can be smaller than that. Sometimes you can find FlexATX boards with AGP and PCI slots -- especially newer models -- but for the most part you're supposed to use the onboard peripherals. A FlexATX motherboard has a very low power requirement; you can generally use a small formfactor (SFX) power supply as low as 180W and have no trouble under most conditions.
NLX: NLX motherboards are anywhere from 9"x13.6" to 8"x10". The I/O connectors are all flat along the left side of the motherboard and there is a slot for a riser card, on which lie your peripheral card slots. The NLX design is generally used in desktop-style cases and requires a special formfactor chassis to mount properly. This is not an ATX derivative in terms of board design, but the power standard is still ATX. There are NLX-compliant power supplies, but they are generally quite rare and the only real differences between an NLX and an ATX power supply are the optional six-pin power connector for FireWire power, and a larger vent in the back of the supply (an NLX power supply also serves as a case fan, pulling warm air out of the machine).
Mini-ITX: This standard was invented by VIA and is maintained by them. It refers to VIA's motherboards that use low-power, integrated x86-compatible processors. Mini-ITX motherboards are extremely small -- 6.7" square, yet they still conform to the FlexATX mounting and power standards. Some Mini-ITX motherboards can run off of AC adapters. The catch? They aren't very fast and they cannot be upgraded.
SFX and TFX: Both are power supply designs for small formfactor machines that use FlexATX and microATX motherboards. They are basically standard ATX power supplies except they don't have the -5V rail, are smaller (TFX is the smallest) and considerably less noisy. As a consequence they offer less power: right now the most powerful small formfactor power supply is 240W.
Desktop: This design was much more popular in the Baby-AT era than it is today, but there are still desktop systems in use out there. Many use an NLX formfactor motherboard; Compaq and Packard Bell sold a great many Pentium-class systems using this design. Usually they have two or three 5.25" drive bays, one (sometimes integrated) 3.25" bay, and internal space for one or two hard drives. It is entirely possible to design a full ATX desktop case, but most of them these days are mini-desktops with small formfactor designs (see below).
Tower: There is no standard for towers, so what one manufacturer calls a mid-tower, another may call a super tower. They can have any number of drive bays (usually between five and twelve) and generally come with a power supply already installed. It's become quite popular to put an acrylic window in the side of tower cases, but this practice can cause problems with the design of the system that force it into non-compliance with FCC regulations (it can cause radio interference). If you see a pre-modded case that does not come with a power supply in it, chances are it is not FCC compliant and you should probably stay away from it. Tower cases can be made of a variety of materials, but steel and aluminum are the most popular. Steel is obviously a lot more durable, but also heavier. Aluminum is flimsy and easily bent and it cannot be easily painted, but it is extremely light and better at conducting heat conductor than steel (which means nothing to the everyday desktop user).
Mini-PC: A relatively new design, the mini-PC (or "cube" system) was introduced by Shuttle come years ago, but now just about every motherboard manufacturer has jumped on the mini-PC bandwagon. Mini-PCs generally have one 5.25" and one 3.25" drive bay and one space for an internal hard drive. The type of motherboard used is the FlexATX, but most mini-PCs come as barebones units -- motherboard, power supply and chassis in one assembled unit. They're light, portable, extremely quiet, and in many instances they are just as powerful as a standard desktop system. The drawbacks are the limited upgrade potential, and limitations imposed by a small power supply (high-powered video cards can have a lot of trouble in mini-PCs).
Mini-desktop: Mini-desktop cases can house Mini-ITX, FlexATX and microATX motherboards. There are a wide variety of mini-desktop designs, some with built-in optical and floppy drives. Nearly every major motherboard manufacturer has at least one mini-desktop system on the market right now. The microATX mini-desktops (and some others) can only use half-height peripheral cards, so be careful what you buy if you're planning on building a system around this formfactor.
Small formfactor: Refers to anything smaller than the ATX and desktop formfactors.
Slot 1 and Slot A: Sometimes called SECC (Single-Edge Contact Cartridge) or SEPP (Single-Edge Processor Package), slot processors were introduced with the Pentium2 and were phased out just above 1Ghz in the Pentium3 line. AMD copied Intel's slot design and called it the Slot A; it was not terribly popular and was quickly phased out. The first Xeon processors were of a slot design, but they too jumped over to socket designs after the first few models.
Socket 370: The old Pentium3 socket design. This encompasses the Tualitin and Coppermine core P3 socket processors. The design was discontinued when the Socket 423 Pentium4 was introduced. Some were PPGA, some were FCPGA, and some were FCPGA2 (see below).
Socket A: All of the AMD Athlon Thunderbird, Duron, and XP processors are of the Socket A design. It's an FCPGA CPU (see below).
Socket 423: Intel's first Pentium4 design. It wasn't terribly efficient and it was discontinued at 2ghz, in favor of the more robust Socket 478 design.
Socket 478: Also called the mPGA-478, this was Intel's last Pentium4 design in the FCPGA2 package. The first core was called the Willamette and it had 256k of on-die cache memory. The Celeron has 128k cache, the newer Northwood core has 512k on-die cache, and the Prescott core can have 1MB of cache. Early models had a 400Mhz frontside bus, and later models offered 533Mhz and 800Mhz FSBs.
Socket 754: The first Athlon 64 processors had integrated heatsinks like the Socket 478 Pentium4 CPUs, and came in a 754-pin package. This model was phased out in favor of Socket 939.
Socket 940: AMD Opteron processors are all of this pin grid array design.
Socket 939: This is the more mature Athlon 64 design. It also covers the Athlon 64 FX and the Athlon 64 X2.
LGA775: Land Grid Array processors have contacts instead of pins. Intel's second generation Prescott core introduced the LGA775 design, and it continues through the latest Pentium4 and Xeon models.
PPGA: Plastic Pin Grid Array. This is a socket processor that has its die on the bottom of the CPU.
FCPGA: Flip-Chip Pin Grid Array. This is what you call a socket processor that has its die on the top of the CPU. FCPGA processors look like this.
FCPGA2: Some Pentium3 and all Socket 478 Pentium4 processors are of this design. It is basically an FCPGA processor with an integrated heatspreader on top of it. It looks like this.
3.5": This is the smaller drive bay that you generally use for floppy drives. Most tower systems have one external 3.5" bay and several internal bays. All desktop hard drives and 1.44MB floppy drives are 3.5". Zip and Jazz drives are also 3.5", and most of the all-in-one media card readers are 3.5".
5.25": This is the larger type of external bay and it's where you usually put your CDROM. All optical drives (CD and DVD) are 5.25", as are tape drives, the SoundBlaster LiveDrive and other frontpanel devices, and one single brand of hard drive: the Quantum Bigfoot, which is long out of production.
9.5mm: Modern notebook computers use drives of this size.