If you’re having to do a complete restore, perhaps because your hard drive has died or you’re reinstalling the operating system, then ideally you’ve got a solid backup system, and you can simply put the catalog, photos, and other files back exactly where they were, double-click on the catalog (*.lrcat file) to open it, and carry on working as before. If you’re trying to restore from backups because you’re moving to a new computer, these instructions go into more detail.
If you don’t have quite such a straightforward restore option, for one reason or another, then you’ll need to restore as much as you do have backed up. Copy (not move!) the catalog and photos back to your computer from the backup drives, keeping an identical folder hierarchy for the photos if you possibly can. For example, if your photos were in dated folders, within year folders, and you put them back in a folder called Pictures, with subfolders for different people’s names, you’re going to have a long job ahead.
Now double-click on the catalog (*.lrcat file) to open it, and survey the damage! Are there question marks on any of the folders or photos? If so, those photos are no longer in the location that Lightroom is expecting, and you need to update the links to point to the new location. Once you’ve finished relinking any missing files, you can then get back to working normally.
If you’re reading this before your hard drive dies, prevention is better than cure, so now is an excellent time to make sure that your backup system would be easy to restore in the event of a disaster.