If you have a problem already, you are not out of luck yet. Many problems can be fixed with a little bit of work and the right software. Some problems are bad enough, that I would recommend a re-install of your operating system. If you develop a problem with your computer and you haven't performed a re-install already, then you should do so. If you are unsure about how to do so, please read this tutorial.
If you have a different problem, or are unsure what is wrong, let's take a look at some of the best tools available to fix your computers ailments. Are you ready to dive in? Grab your suit and let's go!

Hard Drive

If you have a major hard drive problem, then there is only one place to go for help: a program called Spinright ($89.00). This program works across all platforms, and it digs down to the lowest levels of your file system to find the problems with your hard drive. This program truly does pull of miracles in the hard drive department. I highly recommend this program, as it can save your data, and save your drive. Be warned: because it operates on such a low-level in the file system, it can take a long time to go through your hard drive and fix it.
Have you lost a file? Is it not in your recycle bin? The key here is to stop what you are doing! You want to stop downloading things and saving things, as your goal has now become to not overwrite the file. Even though it is gone, it merely pulled a Houdini on you. If you download Recuva(free), you can run this program and it can recover any file that hasn't been overwritten already.

Memory

When it comes to software to check for memory errors there is really only two programs available. One is commercial: Memtest ($12.95) and one is free Memtest86. The important thing with RAM is to make sure to use name-brand RAM. Name-brand RAM shouldn't fail under normal operating conditions if it works in the beginning. RAM issues are usually related to physical issues (a bad stick of RAM, a stick of RAM fails), moreso than anything else. It can have issues however, and these programs will check them every aspect of your memory.

General Computing Issues

Want to know what your computer is doing? Sure, the task manager gives you a solid overview, but if you really want to know, then you need to download Process Monitor from Sysinternals. Process Monitor is an advanced monitoring tool for Windows that shows real-time file system, Registry and process/thread activity. It combines the features of two legacy Sysinternals utilities, Filemon and Regmon, and adds an extensive list of enhancements including rich and non-destructive filtering, comprehensive event properties such session IDs and user names, reliable process information, full thread stacks with integrated symbol support for each operation, simultaneous logging to a file, and much more. Its uniquely powerful features will make Process Monitor a core utility in your system troubleshooting and malware hunting toolkit.

Have you ever wondered which program has a particular file or directory open? With Process Explorer, you can find out. Process Explorer shows you information about which handles and DLLs processes have opened or loaded.

The best methodology is to use these tools to figure out what is wrong and then to use google to troubleshoot the error. You can be sure that someone else out there has had the exact same problem as you. If you get stuck, then feel free to reach out to your favorite geek. And then give them a hug. It just feels good.