The Forbidden Hardware
If you read the first article of this series, you should have a good idea of what is inside your computer. Assuming you have that information, you now need to figure out if the hardware you have discovered in your computer is worth bothering with. Telling you what is worth an upgrade is impractical, as most hardware built in the last few years will provide an upgrade opportunity. Instead, I'll simply tell you what hardware that should stop you in your tracks, should you find it in your PC.
Motherboards Lacking a Modern CPU Socket - If your motherboard has AMD's 462(A), 939, 940, or 754 socket, or if it has Intel's 370, 423, or 478 socket, then your motherboard isn't compatible with the latest processors and it therefore should be replaced. Your processor will also need replaced. And it is a fair bet your previous RAM isn't going to be useful in modern PCs, so that also will likely need replacing.
Motherboards Lacking Expansion Capabilities - If you're upgrading a pre-built PC, you may find yourself stuck with a motherboard that only has enough connectors for the hardware your computer originally shipped with. If the provided motherboard doesn't have free PCI, SATA, or RAM slots, then you may need to buy a new motherboard, at which point you'd also have to go through the process of making sure your new motherboard will function correctly with your older hardware.
Motherboards Lacking BIOs Support of New Processors - Even if you've determined that your motherboard can physically accept a certain processor, you're not quite out of the woods yet. Not all motherboards that support a certain socket support all processors that can fit that socket. For example, budget AM2+ boards do not necessarily accept the highest-end Phenoms, and budget Intel motherboards that accept Core 2 processors do not always accept the latest Core 2 processors. To find your what your motherboard is compatible with, visit the website of your motherboard manufacturer.
As you've probably noticed, these are all motherboard issues. This is because, should you need to upgrade your motherboard, you are effectively re-building your PC. You will have to not only take apart most of your system, but you will also have to verify that your new motherboard is compatible with your older hardware and will fit well in your case. You will also have to reinstall your operating system. This isn't necessarily a bad idea, but doing so means you are re-building your system, and is outside the scope of this guide.