Frankly the easiest way to find out what you can use is to turn it off and open it up. For a desktop the hardest part is usually opening the case; try not to break any plastic tabs. If necessary, to open your case, check with the manufacturer.
You need to determine if you are using SATA or IDE (or, unlikely but maybe in very old computers, SCSI). So first check for IDE or SATA before you buy, IDE has a wide 3-inch ribbon cable while SATA uses a small 1/4-inch cable. If you have both then I would go with SATA as this seems to be the most popular now.

1. You need to make sure you have a drive bay available. 2. A SATA or IDE slot on the motherboard (IDE ribbon cables can support two devices so you may also be able to chain it there if your DVD is on a ribbon by itself). 3. You need an available power connection that is appropriate: molex (for IDE) or SATA power. Y splitters and molex to SATA power converters are available if needed.
You also have to decide how fast you want it to rotate, 5400 RPM are slower, 10000RPM are expensive and often have special/extra cooling needs, so I go with 7200RPM. Then decide how much space you need; at today's prices you would probably go with 1TB or 2TB. If you buy an OEM SATA harddrive it typically does NOT include the SATA cable so you will need one of those. And buy a power splitter or converter as needed.
Slip the drive in, screw it down, attach power, attach the ribbon or SATA cable, and power up.
Always ground yourself before reaching into any computer even though it is off; as the static electricity can damage components. As long as you do not stick your hand inside while it is on, it is safe to test it with the case open. Once you know it works, power down and put the computer back together.