The motherboard (also commonly referred to as mainboard or mobo), is the one peice of hardware that connects to everything else. It's important to know what sort of things a mainboard has.
The most obvious is the the CPU socket. This is where the CPU goes. There are PGA and LGA sockets. Currently, Socket AM2 is used for AMD Sempron, Athlon and Phenom series CPUs. It's a PGA socket because the pins are on the CPU and they plug into the mainboard. Intel on the other hand uses both LGA 775 and LGA 1366 sockets for their systems. LGA simply means that the pins are on the mainboard instead of the CPU. Socket LGA 775 is for later Pentium 4 and Celeron CPUs, and was used exclusively for the Pentium D, Core, and Core2 series CPUs. LGA 1366 is used exclusively for Intel's Core i7 CPUs.
The next most obvious attribute to any mainboard is the chipset. The chipset is the one chip that manages everyhing on the mainboard. There are two pieces to the chipset: the northbridge and southbridge. The Northbridge handles the front-side-bus (FSB) which is the communication channel between the CPU and the memory, and also controls the expansion slots, like AGP, PCIe, PCIx, and PCI. When applicable, the northbridge also manages the onboard video on mainboards that have them. The Southbridge manages the chips that control the ethernet, USB, firewire, and driver controllers. Some mid-range boards and up have two drive controllers for more SATA ports and USB ports. AMD, Intel, Nvidia, SiS and VIA all make chipsets for common desktop mainboards. The SiS and VIA chipsets are usually found on cheaper boards, but in many cases, they work just fine for most basic uses. They don't overclock, they usually don't have any special features, and they almost always have integrated graphics. Intel and AMD also have some simpler chipsets.
Some chipsets are just more popular than others. Since they control the major aspects of the mainboard, they often have to be taken into consideration when choosing a mainboard. It's quite common today for a mainboard to have the ability for dual video cards. These technologies are called SLI from Nvidia and Crossfire from ATI. To use these technologies, a mainboard would need at least 2 16x PCIe slots, and a chipset that supports it. Intel 965, 975, P35, P45, X38, X48, X58 chipsets all support ATI's Crossfire technology, as do AMD 780 and 790 series chipsets. If you're looking for SLI, your only options are Nvidia's 590 SLI, 680i, 780i, 790i, and Intel X58 chipests. Intel's P45 chips are very popular for the Core2 crowd, while X58 is the only options for Core i7 adopters. For AMD, a lot of power-house rigs are using 790 series chipsets, while many building budget gaming rigs, or HTPCs are using AMD's 780 chipset. The 780G is one of the most popular due to it's low-power design and it's excellent performance for onboard video. I've heard it can do HD-DVD/BluRay playback.
There are several different types of expansion slots that can be found on a mainboard, but only a few actually show up in common consumer boards. PCIe slots are specified by the number of data lanes they utilize. For a graphics card, you'd more than likely be using a 16X PCIe slot. 4x and 8x PCIe slots are more common on Server and Workstation type boards, but they can be found. These are most often for RAID controllers. 1x PCIe slots are common, but the cards the use them aren't. It's gotten much better over the least few years, but the only type of card you'll find for 1x PCIe slots are a handful of sound cards, the occasional basic video card, and USB/Firewire cards.
Most mainboards follow a simple rule that any consumer must remember: You get what you pay for. If you want more of this, or more that, you'll probably find it on the more expensive boards. Basic boards will usually have onboard video (along with nearly everything else), a single 16x PCIe slot, single 1x PCIe slot, a couple legacy PCI slots, 2 to 4 SATA ports, 1 IDE channel, 1 Floppy port, 2 DIMM slots for memory, and the usual assortment of power plugs. The features get better the more expensive you get, although only up to a certain point. For AM2 or LGA 775, more than $150 and more than $250 for LGA 1366 boards, and your extra money starts to lose it's worth fast. A few more lights and SATA ports isn't very worth an extra $50 to most people.
Mainboard brands also have important roles to play. Some brands are just better than others. This isn't a definitive "who's crap and who's not" list, but I'm going by the general consensus. ASUS, Gigabyte, EVGA, MSI, Intel and DFI are respected makers in the mainboard industry. On the lower end of the totem pole, you'll find Asrock, Biostar, Foxconn, Jetway, PCChips, and ECS. The cheaper boards will try to lure you with impressive prices, but remember: you get what you pay for.
As for CPUs, Intel and AMD have brought us a great selection. Intel holds the performance crown and has since their Core2 chips came on the market. AMD still makes one helluva chip and offer some great bang-for-your-buck deals.
AMD chips are most common for the basic builds, or for someone's who's on a tight budget. For a basic computer, AMD's most popular chip is their Athlon X2 4850e. This dual core CPU has a clock speed for 2.5GHz, 512Kb L2 cache per core, and is very efficient in terms of power usage. It's often paired with an AMD 780G-based mainboard. This chip is perfect for a small home-server and low-end High Definition HTPC setups too.
For the power-user on a tight budget's AMD's new PhenomII 920 and 940 make excellent quad-core CPUs for the price. They beat out Intel's Q6600 in benchmarks and will run most games just fine.
There are more recommendations for Intel CPUs because you'll get better performance with Intel, in nearly every type of build. AMD has some great prices, which is why I nominated them for a good basic build. For most situations, an Intel setup will be much better performing.
For a mid-range CPU that both power users and gamers can appreciate is Intel's Core2 Duo E8xxx series. The E8400 and E8500 are very similar in both price and specs. Their higher clock speed and extra 6Mb L2 cache is a big help for gaming, and their incredibly fast in most other tasks. They're also a great deal at around $170 currently.
For the high-end few holds barred, rig, Intel's Core i7 CPUs should be a no-brainer, the Core i7 920 in particular. The 920 is a very impressive overclocker, isn't overly warm so it's not a pain to keep cool, and give the faster 940 and 965EE chips a good run for their money. If need or like to have your rig doing a lot, all the time, then a quad-core is a good choice. And for quad-cores, you can't do better than an Intel Core i7. They can be a bit on the expensive side, but most of the cost is in the mainboards and the fact that they only utilize DDR3 RAM, which isn't nearly as cheap as DDR2.