How should I go about mounting an external hard drive without an enclosure?
I have a IDE/SATA to USB adapter. But where should I put the drive, what side should be facing upwards, etc?
Whenever I have to use an internal hard drive outside the case, I usually just lay it on the metal side with the components facing upwards. As long as this isn't a permanent solution, you should be fine. If you plan to leave the hard drive out in the open all the time, then I would highly recommend an enclosure
Another concern with a naked hard drive is picking it up and moving it while the heads are not parked. A sharp bump or drop of the hard drive could cause the head to crash into the platter, causing permanent damage and data loss. Modern laptop hard drives sometimes have accelerometers that sense the hard drive is falling and park the head before it makes impact, but I'm not sure if desktop hard drives commonly have this or not. You can read up about head crashes on Wikipedia if you're interested.
Below is my original answer, which I'll just leave for context.
If you have an internal hard drive that you want to connect to a computer without opening the case (or if it's a laptop), then there are USB adapters you can purchase, such as this one that adapts both PATA and SATA to USB, which I have used at work with no problems.
Another solution is the Thermaltake BlacX SATA docking station. I have two of these for my own purposes and they work great as well. I use these for a rotating backup system so I can always keep one hard drive off-site and hot-swap them as necessary.
I used to work in a lab where we had dozens of naked hard drives at any point. Data restoration is expensive (if not the hard drives..any more), so just a couple more things to keep in mind: