Welcome to the forum.
Now, please tell us exactly what you are trying to do. When you choose Save, Save_As, or Save_As_a_Copy, your "video" is not being Saved - only the Project file, .PREL, is being Saved.
To get your Video to your computer, you need to use Export, or Share, depending on exactly what you need.
Also, please give the exact wording on the error message. This could well be indicitive of not choosing the correct button, or could mean there is something going on with your computer.
Are you trying to Save to a networkded drive? Are you logged-in as an Administrator? Could be various problems, but the exact error message might tell us something.
Just some supplemental questions since you were a little short on details:
1. What version of Premiere Elements are you using?
2. Have you taken a hard look at your computer resources, such as, available RAM and free external hard drive space as well as free local hard drive space? Have you defragmented your computer recently? Where are you trying to save this export to and is there enough free space there?
a. look to wherever you have the scratch files directed to see if you are piling up preview files there (Edit Menu/Preferences/Scratch Disks)
b. Unless you have made changes, the default location for the Adobe Folder is My Documents/Adobe/Premiere Elements. Besides checking for pile ups of preview files, look in the AutoSave Folder there and see if you have a pile up of AutoSaves because you had Edit Menu/Preferences/AutoSave on and the way you had it set for save every and maximum projects.
c. How long have you been using the program and is this the first and only video that has given you this problem?
d. Just what length and format of export are we talking about here?