LAST UPDATED: January 7, 2014
Haswell generation (4th gen), while older generations include Ivy Bridge (3rd gen) and Sandy Bridge (2nd gen).
Sandy Bridge processors have model numbers in the 2000's, such as the Intel Core i5-2500. The built-in graphics cards on Sandy Bridge processors work with Mac OS X Lion and Mountain Lion (but not Snow Leopard), and come in two versions: HD 2000 and HD 3000. Unfortunately, only HD 3000 graphics are officially supported. HD 2000 sort of works, but it doesn't have graphics acceleration, so it's recommended that you just buy a separate ("discrete") graphics card for your computer instead.
Nowadays, most Hackintosh builds use Ivy Bridge. Ivy Bridge processors have a model number in the 3000's, such as the Core i5-3450. Mac OS X Lion 10.7.5 and all versions of OS X Mountain Lion support Ivy Bridge natively. Snow Leopard doesn't support Ivy Bridge at all, though you might be able to get it working with iBoot Ivy Bridge. HD 4000 graphics, which is built into some Ivy Bridge processors, works with OS X Mountain Lion and Mac OS X Lion version 10.7.5 (and newer). Intel HD 2500, the successor to Intel HD 2000, works with OS X Mountain Lion version 10.8.3 and (newer).
The Haswell generation of Intel Core processors is supported natively by OS X Mountain Lion 10.8.5 and all versions of OS X Mavericks and Yosemite. You can probably also install Snow Leopard on a Haswell computer by using iBoot Haswell, although it's not recommended as a long-term solution. Haswell processors have a model number in the 4000's, such as the Core i7-4770. They include built-in graphics cards, which come in two versions: HD 4600 and HD 4400. HD 4600 graphics work with OS X Yosemite, Mavericks, and Mountain Lion (version 10.8.5 and above). HD 4400 does not yet work in Mac OS X.
The highest end Intel Core processors are known as "Sandy Bridge-E", "Ivy Bridge-E", or "Haswell-E". Sandy Bridge-E processors have model numbers in the 3800's and 3900's, such as the Intel Core i7-3820. Meanwhile, Ivy Bridge-E processors have model numbers in the 4800's and 4900's, and Haswell-E has numbers in the 5800s and 5900s. Mac OS X does not completely support these processors, because they use a different socket from the rest of the Intel Core line. CPU power management and sleep mode only work for Ivy Bridge-E processors in OS X 10.9.2 and newer. And even then, certain workarounds are still required. Sleep mode and CPU power management are only guaranteed to work for Sandy Bridge-E processors in OS X 10.9.0 and newer. Haswell-E processors aren't officially supported in any version of Mac OS X at all (which means no sleep or CPU power management), but you can still get those processors to work in OS X 10.9.4 (and newer) with a few extra steps.
P.S. Intel Sandy Bridge processors don't work very well with Mac OS X Snow Leopard version 10.6.8. I recommend that you update to version 10.6.7 instead. You can still update to Mac OS X Lion from 10.6.7 (updating to Mountain Lion requires 10.6.8, but you might be able to circumvent this requirement by spoofing your system version).
Haswell generation) are not supported in either OS X Mavericks or Yosemite. While there may be workarounds, in general, it's a no-go right now.
However, if you're putting together a budget Hackintosh, then using one of the older-generation Pentium or Celeron processors isn't necessarily a bad idea. These processors use mostly the same underlying architecture as their higher-end Intel Core counterparts. For instance, the Intel Celeron G530 uses the LGA1155 socket, which is also used by Intel Core processors in the Sandy Bridge and Ivy Bridge generations. As a result, the G530 works very well with Mac OS X.
However, Mac OS X compatibility for extremely old Pentium and Celeron processors (think pre-2010) will vary widely, since most of these processors use different CPU sockets. In some cases, it comes down to which motherboard you're using. In other cases, the processors themselves may simply be too old. For instance, Pentium M processors produced before 2008 are 32-bit CPUs, meaning that they don't work with Mac OS X Lion or Mountain Lion, which are 64-bit operating systems.
Unlike the integrated graphics cards found in Intel Core processors, the integrated graphics cards in Pentium and Celeron processors don't work with Mac OS X. This includes Intel's GMA series of integrated cards. You can install makeshift kexts from OSx86.net that allow Intel GMA to display Mac OS X at higher resolutions, but there's no way to enable graphics acceleration. Instead of using integrated graphics, you will have to buy a separate graphics card for your Hackintosh.
Like Sandy Bridge-E and Ivy Bridge-E, Intel Xeon is not completely supported in Mac OS X because it uses different sockets from the mainstream line of Intel Core processors. While you can still boot Mac OS X from a computer using an Intel Xeon processor, sleep mode and CPU power management will not work. You'll also probably have to perform a few extra steps during the installation process, in order to get everything working properly. If you don't mind that, then the next big challenge is finding a good motherboard. Since Intel Xeon processors are designed for servers, and few people ever bother installing Mac OS X on a server, there are very few successful Xeon Hackintosh builds that can set an example for future builders.
Our advice? Unless you really need the extra cores in Intel Xeon processors, just buy a high-end Intel Core processor for your Hackintosh instead.
Technically, you can install most versions of Mac OS X on an AMD Hackintosh. You just need an AMD kernel-- a modified version of the vanilla kernel that works with AMD processors. However, using modified kernels is a tricky business, and no matter how you spin it, an AMD Hackintosh will never be as stable as its Intel counterparts.
Even if you get Mac OS X to install, that's only the first step. In fact, the subject of setting up an AMD Hackintosh deserves a whole other website by itself. To put it one way, unless you're a fan of self-flagellation, you should try to avoid AMD altogether.
The Haswell generation of Intel Core processors is a great choice for any Hackintosh, especially with the newest versions of Mac OS X. If you're looking for something really cheap, the newest Intel Pentium and Celeron processors work very well with Mac OS X, as well.
However, try to avoid Intel Xeon, Sandy Bridge-E, Ivy Bridge-E, or Haswell-E processors, unless you're okay with some extra work (and the possible need for experimentation). Most of the time, simply buying a mainstream Intel Core processor will be a safer decision. In addition, AMD processors are a pain to work with on Mac OS X, and should be avoided whenever possible.