Circuit board on the bottom. Make sure that whatever surface you put it on is non-conductive. If necessary use a few sheets of newspaper, a magazine or something like that under it.
Be careful of how "off" your computer is when it appears to be switched off. Newer boxes are software controlled and even though they appear to be off are still live at some point. eg To turn mine completely off, I have to push and hold the power button for 10 seconds - a normal push and release only suspends the computer, leaving it partially powered for a quick resume. Best to unplug it from the power supply completely. (There is room for discussion here re earthing back to the power point, but it depends on the electrical safety standards of your country. I don't know yours, so I'll skip it.)
Oh, and before you handle any part of your computer internals or your new drive, ground yourself and your electronics. I work on the kitchen bench near the sink and regularly ground everything by touching the sink first, then both the gear (eg the casing) and the sink simultaneously. I also wear bare feet - in a home environment socks and synthetic soled shoes can cause static build up in your body which can fry any electronics you touch. You can buy earthing wrist straps and other stuff, but for a one-off at home like you're doing I don't think it's necessary.
(Edited to add bits as I thought of them
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AA