This is intended to be a one-stop shopping spot for testing your hard drive outside of Windows.  If you're able to get into Windows, some of the manufacturer's offer tests that will work inside of Windows (specific link for SeaTools for Windows in the SeaGate section).   

Suggestions are welcomed!

What you'll need:
- a blank CD
- a computer with an internet connection where you can download a file, install a small program, and burn a CD (doesn't have to be the computer that you'll be testing).

1.  Determine the make and model of your hard drive(s)

- In Windows XP - Go to Start...Run...and type in "devmgmt.msc" (without the quotes) and press Enter.  Click on the + sign next to Disk Drives and use Google to search the information listed there).  Also check for USB, eSATA and 1394 devices.  In short, you're testing any and all storage devices (remove any flash drives)

- In Windows VISTA/Windows 7 - Go to Start...and type in "devmgmt.msc" (without the quotes) and press Enter.  Click on the + sign next to Disk Drives and use Google to search the information listed there.  Also check for USB, eSATA, and 1394 devices.  In short, you're testing any and all storage devices (remove any flash drives)

- In Windows 8 - press the "Win" key and the "X" key at the same time.  Then, when the menu pops up in the lower left corner of the screen, press the "M" key - that will open Device Manager.    Then click on the > sign next to Disk Drives and use Google to search the information listed there.  Also check for USB, eSATA, and 1394 devices.  In short, you're testing any and all storage devices (remove any flash drives)

- From outside of Windows: Unplug the system from the wall, remove the battery (if it's a laptop) and crack the case open.  Look inside to locate the hard drive(s) and write down the make and model number (although the model number isn't required for the test, it will help you to identify what sort of drive to replace if the test says that the drive is bad).  Also check for USB and 1394 devices.  In short, you're testing any and all storage devices (remove any flash drives)

NOTE:  It's strongly recommended that you use the test from the manufacturer of your hard drive.  Most manufacturers build in tests that are only for their drives (in addition to the standard S.M.A.R.T. tests) - so using their test will give more accurate results.

NOTE:   Please run the test on all hard drives attached to your system.  This is necessary as a fault in any of the drives can cause problems with the entire system.

2. Locate the support website of the hard drive manufacturer from the following list and download the appropriate file to your computer.  FWIW - the first link for each drive usually contains more information/instructions about running the test.  If you cannot locate a test for your hard drive - run the Hitachi DFT (for older systems designed for IDE drives - primarily XP and earlier) or the Seagate SeaTools (for newer systems designed for SATA drives - primarily Vista and later).

NOTE:  If using an OEM Win8 installation, it may be difficult to boot from CD/DVD due to Secure Boot.  If so, try running the appropriate Windows version of the test(s).

Hitachi - http://www.hitachigst.com/hdd/support/download.htm#DFT  Download this image for CD's: http://www.hitachigst.com/hdd/support/downloads/dft32_v416_b00.iso 
FYI - I have seen issues with v415 on some Asus laptops - it will erase the boot settings, requiring you to go into the BIOS and reset the BIOS to default values.

Western Digital  -  I've had issues with running this from a CD/DVD - try the Hitachi DFT (on older/IDE systems) or the Seagate SeaTools (on newer/SATA systems) - http://support.wdc.com/product/download.asp?level1=6&lang=en  Download the Diag504fCD.iso file on this page:  http://support.wdc.com/product/download.asp?groupid=601&sid=30&lang=en
The stored copies that I had linked to have been removed from my hosting accounts.  As such, I have removed the links those copies that I had stored.

Seagate - http://knowledge.seagate.com/articles/en_US/FAQ/201271en (using the Graphical Interface of Seatools for DOS).  Download the Seatools for DOS ISO CD-ROM Image from this link:  http://www.seagate.com/support/internal-hard-drives/consumer-electronics/ld25-series/seatools-dos-master/

Maxtor - Seagate SeaTools has replaced the Maxtor diagnostics.  Use the Seagate SeaTools links above.

Toshiba - http://storage.toshiba.com/storage-services-support/warranty-support/software-utilities

Fujitsu - sold to Toshiba on 01 Oct 2009.  No tests available AFAIK.  Use the SeaGate SeaTools for Vista and newer systems, use Hitachi DFT for XP and older systems.  Thanks to MvdB:  You can find RMA info, warranty info AND diagnostics for the Fujitsu drives here: http://sdd.toshiba.com/main.aspx?Path=ServicesandSupport/WarrantySupport/SoftwareUtilities#diagnostic

Samsung - Go to http://www.samsung.com/us/support/ and select "Hard Disk Drives" under the Office category.
Enter the information about your hard drive model, then click on the Get Downloads link on the next page, then on the Software link on the following page.

If there's no downloads available, try this link - it seems to be the same version (ESTOOL v3.01) for every drive: http://org.downloadcenter.samsung.com/downloadfile/ContentsFile.aspx?CDSite=US&CttFileID=220644&CDCttType=SW&ModelType=C&ModelName=HA250JC&VPath=SW/201011/20101118113245515/ESTOOL301v.iso

You must unzip the file before you can burn (NOT copy) the ISO image to a CD.

IBM - IBM has merged with Hitachi.  Use the Hitachi DFT (Drive Fitness Test) listed above.

3.  If you do not have a disk burning program that's capable of burning ISO images to your CD drive, I suggest this free tool:  http://isorecorder.alexfeinman.com/isorecorder.htm  Select the appropriate version by following the instructions, Install the program, then reboot your system.
FYI - an ISO burner will place files in a certain order on the disk, this is what makes the disk bootable.  You CANNOT  just copy the file to a CD - it won't work!

4.  Insert the blank CD into the CD drive and dismiss any dialogs that may pop up.  Then right click on the ISO file that you downloaded and select "Copy Image to CD" (it should be the top choice on the context (right click) menu.  Follow the prompts to record the image.  Remember that the slower the speed you select, the less likely it is that there will be errors in the burn process.  Most disk images are small, so the process shouldn't take very long even at the slowest speed.

5.  Boot from the CD to run the test.  Here's a partial list of ways to access the built in boot menus of different systems.  Press the appropriate key when the first screen shows up after you turn the computer on.

HP/Compaq - Esc (starting to use F9 also) (F10 will usually access BIOS)
Dell - F12  (F2 will access BIOS)
Gateway/eMachines - F10 (moving to F12 with newer models)  (F2 will access BIOS)
Acer - F12 (not always)  (F2 will access BIOS)
Toshiba - F12 (older systems will be different)  
Sony - usually will boot to CD, may ask you to "Press any key to boot from CD"
Asus - Esc (F2 will access BIOS)
Samsung -  Esc (F2 will access BIOS)
Nokia - F12  (F2 will access BIOS)
Lenovo - F12 (?Del wil access BIOS)

When in doubt, either access the BIOS (Setup) and make the CD drive the first boot device (before the hard drive), or post in the forums for assistance.

6.  Follow the prompts on the screen to run the test.  In particular run the Long/Extended/Complete/Comprehensive/etc test - as the shorter tests are not completely reliable.  

My greatest experience is with the Hitachi DFT - and you can just press Enter (2 times) until you get to the screen where it asks you to agree to the conditions (press A), confirm the devices listed (press Y), select Advanced (press A), then Start (press S) there.
For the Seagate SeaTools - boot from the CD, click on "Accept", wait for it to finish detecting the drives, then in the upper left corner select "Basic Tests", then select "Long Test" and let it run.
NOTE:  If you have more than one drive installed, run the test on each drive.

If you encounter any problems running the test - just post them to the forums and someone will be glad to help you out.

Maxtor - Seagate SeaTools has replaced the Maxtor diagnostics.  Use the Seagate SeaTools links above.

Toshiba - http://storage.toshiba.com/storage-services-support/warranty-support/software-utilities

Fujitsu - sold to Toshiba on 01 Oct 2009.  No tests available AFAIK.  Use the SeaGate SeaTools for Vista and newer systems, use Hitachi DFT for XP and older systems.  Thanks to MvdB:  You can find RMA info, warranty info AND diagnostics for the Fujitsu drives here: http://sdd.toshiba.com/main.aspx?Path=ServicesandSupport/WarrantySupport/SoftwareUtilities#diagnostic

Samsung - http://drive.seagate.com/content/samsung-en-us  AFAIK, no Windows diagnostic tests are offered for these drives through Seagate

IBM - IBM has merged with Hitachi.  Use the Hitachi Windows DFT (Drive Fitness Test) listed above.

All SSD's - generally not available.  If available, it'll be located somewhere on the SSD manufacturer's support website.

Generic Testers:
Crystal Mark (free) - primarily for testing SSD's (actually a benchmarking software):  http://crystalmark.info/software/index-e.html

HDTune (free) - primarily for testing HDD's (just plain HD Tune, NOT HD Tune Pro):  http://www.hdtune.com/download.html