install OS X Lion or Mountain Lion on another hard drive

Perform the installation normally, but when given the choice of picking a destination, specify the second drive or partition. The installer won't be automatically deleted if this is done.

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Just as you install OS X on your internal hard drive. First of all, make sure that the hard drive where you want to install OS X is formatted with "Mac OS Extended (Journaled)" and uses GPT. To format the hard drive, read > http://pondini.org/OSX/DU1.html

If you are already using Mountain Lion, just start in OS X Recovery holding Command and R keys while your Mac is starting. Then, select the option to reinstall OS X and choose a different hard drive as the destination to install OS X. Note that this will change the startup disk so, to start in your internal disk again, hold the Option (Alt) key while your Mac is starting and select your internal disk. Finally, open System Preferences > Startup Disk, and choose your internal disk

Get an external drive equal to the capacity of your current startup drive. Do the following:

Repair the Hard Drive and Permissions - Lion/Mountain Lion

Boot to the Recovery HD:

Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.

Repair

When the recovery menu appears select Disk Utility. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported then click on the Repair Permissions button.

Next, prepare your external drive:

Drive Partition and Format

1. Select your external drive (this is the entry with the mfgr.'s ID and size) from the left side list. Click on the Partition tab in the DU main window.

2. Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Partition button and wait until the process has completed.

Now, make a bootable clone

Clone using Restore Option of Disk Utility

1. Select the destination volume from the left side list.

2. Click on the Restore tab in the DU main window.

3. Select the destination volume from the left side list and drag

it to the Destination entry field.

4. Select the source volume from the left side list and drag it to

the Source entry field.

5. Double-check you got it right, then click on the Restore button.

Destination means the external drive. Source means the internal startup drive.

Late 2012 iMacs and Mac minis use a special OS X version which you can't get from the App Store, so it's normal that you see that OS X isn't compatible. Instead of downloading OS X from the App Store, start in OS X Recovery to install OS X onto the hard disk