No (provided you stick with the same operating system) .. look here:
http://www.jriver.com/purchase.html
To be clear, astromo means that the OSX and Windows versions of MC are licensed separately (if you want to switch to or from a Mac, you need to buy it again, or run it in a VM).
You can certainly switch from OSX 10.7 to 10.9, or Windows XP to Windows 8.1, without a new license, however. The original restore key you generated will almost certainly be expired, but don't worry about that, you can just download the latest version of MC from Interact, install it, and then enter your license key to activate/license the copy. You can do this up to 10 times per year (and if you ever run out of "restores" for a legitimate reason, just come here and explain what happened and someone will help you out).
The Library is the database that MC uses to keep track of all of your files (know where they are on disk, all their tags, cover art, etc).
Anywhere here on the forums or on JRiver's wiki, when you see us refer to "the Library" we are not talking about those AIFF files, we're talking about the little database files MC uses to keep track of everything. That's the part of MC that distinguishes your copy of MC from my copy of MC. They're essentially "pointers" to the files on disk.
What you'll want to do is:
1. On the OLD computer, open MC and make a
Library Backup. This is a ZIP file, and will be pretty small. Copy this ZIP file over to the new computer.
2. Separately, copy the AIFF files over from the old computer to the new computer, taking care to copy them into the same "location" on the new computer as where they were originally on the old one. So, if you originally put them in your Home Directory's Music folder, put them in that same place on the new computer. If you make a new user account, make sure it has the same username (so that the Home Directory path is identical).
3. The above will, obviously, be easiest if you have the audio files on a separate disk which can be moved to the new computer (or even easier if they are on a network disk of some kind). If so, and the secondary drive was called Drive M, or Drive E, make sure it gets the same drive letter on the new computer (OSX will do this automatically, unless the drive's name conflicts with an existing drive name on the new computer). If not, you may be able to remove the hard drive from the old computer (temporarily) and plug it into the new computer. You could also do this over a network if you can sort that out, but it'll be faster (and probably a bit easier) to do it directly.
4. On the NEW computer, open MC and get it registered. Skip importing things for now or setting any of your options, just worry about getting it licensed.
5. Restore the Library Backup you made in step 1 above (using the Library Manager again), and choose to restore both the Library and Settings.
That should be it. You may need to go back into Options and tweak the settings under Audio and Video to be appropriate for your new computer.