While it's not often that hard drives crash or have other problems, it does occasionally happen. When it does, the easiest and recommended way for managing the internal drive is to have a local operating system and local hardware interface with the drive; however, there may be times when troubleshooting and managing the crash requires you to use alternative methods for connecting the drive to a computer.
Over the years, several methods to do this have evolved. In older computers that did not have USB or FireWire, it was common practice to swap hard drives between systems, or put the drives in extra expansion bays. This is still a solution that can work both to retrieve data as well as fix seemingly broken drives. I have been able to restore numerous hard drives that Windows would not format by placing them in my Mac and using Disk Utility to repartition and format them.
Keep in mind that putting potentially faulty hardware in your system may cause hardware hangs and system crashes. Be sure to protect your own data when doing this by unplugging your hard drive first. You can easily install a fresh copy of Mac OS X on an external drive, boot off that drive, and then use your Mac to run diagnostics on a faulty internal drive, thus protecting your internal drive and data while troubleshooting.
Using "Target Disk Mode" is a similar option. With Target Disk Mode, you can connect the computers via FireWire and mount one computer's hard drive on another one. To do this, restart one computer while pressing and holding the "T" key. This will essentially turn the computer into an external hard drive that can be connected to other computers. Keep in mind that while this is convenient, the drive will still be run by the same hardware controllers. If the problem you're troubleshooting is because of the controller, you may still have trouble reading the drive. Also keep in mind that this will only work with computers that have onboard FireWire controllers and connectors.
A second approach is to use an external drive enclosure (kits and options available here). Similar to swapping the drive with another computer, these enclosures offer another controller and firmware to interface the drive with, which may allow it to function at least enough to retrieve data and then reinitialize it. External enclosures can be either FireWire or USB, many of which will fit both laptop and desktop drives. In addition, you can also use a network-attached storage device. However, keep in mind that using a NAS will usually require initializing the drive for the NAS. So this may only be beneficial if you are trying to format a drive that will not format in another system.
The last option is to use an adapter cable such as the NewerTech Universal Drive Adapter. These have small controllers embedded in its FireWire or USB cable that can connect to many types of hard drives. For people who regularly troubleshoot internal hard drives, these cables are convenient.
NOTE: External drive options can be used for any hard drive or storage device that uses an SATA or ATA interface, including external drives for which the drive is not intended to be removed. If the device is broken and you are not interested in pursuing a warranty replacement, you can open it up and remove the drive itself to try using in another enclosure or system.