Mac or Windows? What version?
Which exact NUMBER version of Lightroom?
What kind of "backup" are you opening?
Lightroom will create backups of the catalog file (*.LrCat). Since your images are not "in" the catalog, but are on the hard drive just like any other document, you need to do backups of the images outside of Lightroom using whatever backup method of your choosing.
If LR thinks your images are missing, the common problem is that your used Finder or Explorer to move them or rename a folder. If you did this, just use the LR Folders panel in the Library module to tell LR where you moved things to.
I just opened up a backup from last month, hoping to restore pictures the app says could not be found, and few of the pictures on the backup can be found as well. I wanted to restore what I had to make sure I didn't lose others, and it won't restore my library from the backup I did a few minutes ago. Can someone help me? Why won't the backups restore pictures? I don't want to lose them all!
The photos are not in your catalog files, so restoring a backup of a catalog will not fix the missing photo problem. The photos are stored on your hard disk(s) somewhere, in the place you put them, or in the place you instructed Lightroom to put them.
So to fix this problem, you have to locate the photo on your hard disk, using your operating system's search feature (search by file name) and then you can connect Lightroom to the new location of the files using these instructions: Adobe Lightroom - Find moved or missing files and folders