Where does Photoshop 14 store user data?

The default location for the catalog folder is C:\ProgramData\Adobe\Elements Organizer\Catalogs.  The catalog folder will have the name of the catalog (e.g. My Catalog), and the db file will be inside that folder.

The catalog file is named catalog.pse14db.  If you have copied it to the new computer, simply double-click on it and the old catalog should open.  But as I said, depending on how you have moved the file structure to the new computer, the catalog may show the files as missing.  If you have not yet moved the catalog file structure to the new computer, you will need to also copy the entire folder containing the catalog.pse14db file.

The Organizer stores its information in a catalog folder.  The best way to transfer your catalog to a new computer is to do a backup and restore.   If you are able to do so, you can use the old computer to backup the catalog on an external drive and then restore the catalog on the new computer.  Depending on how you have organized your files on the new and old computers, you may end up with two sets of photos.  Once the restore has been successfully completed, you can delete the files that you simply "moved" to the new computer.  (There are probably ways to avoid this issue.) 

Let us know if the backup/restore option is not available for you.  In that case, you could perhaps copy just the catalog folder to the new computer (if you have not already done so) and you may then need to relink the files.  But first we need to know what your situation is.

For more information about moving files to a new computer and using the backup/restore option, see this HelpX Article.