This is one of the common question asked on Adobe Forums, Elements Village or Feedback.photoshop.com, which is very unfortunate condition to be in. If this has happened with you first time and you were only using Adobe Photoshop Elements, there shouldn't be any problem related to installation of Adobe Photoshop Elements on same machine after restoration of Operating System.
If you don't lie in above scenario and had installed Adobe Photoshop Elements on multiple machines, you may face problem with re-installation on the computer after restoring operating system on this computer. If this is one of the roadblock to you, we recommend to connect with Adobe Customer Support and explain them the scenario. Basically software installed on your machine was not properly deactivated which may create problems in re-activation after OS restoration.
This is one of the basic problem anyone can face, but next part is more important. Before we move to next things to do, please make sure that you have Catalog Backups ready with you and if you never took backup then nothing can be done. If you were able to restore data form your hard drive then all things can be set appropriately which shall be taken up in separate article. Now if you have backups ready with you on any of the external hard-drive, just connect it and go to restore workflow under File Menu.
If all this is not applicable and you were regularly synchronizing your data on www.photoshop.com, you can simply restore all the photographs back in new catalog with appropriate albums and Tags. There is a separate article covering more details about resorting from Photoshop.com through backup/Sync feature of adobe Photoshop Elements.