I\'m making an app that will be installed and run on multiple computers, my target is to make an empty local database file that is installed with the app and when user uses
I'm not seeing anybody suggesting SQL Compact; it's similar to SQLite in that it doesn't require installation and tailors to the low-end database. It grew out of SQL Mobile and as such has a small footprint and limited feature-set, but if you're familiar with Microsoft's SQL offerings it should have some familiarity.
SQL Express is another option, but be aware that it requires a standalone installation and is a bit beefier than you might need for an applciation's local cache. That said it's also quite a bit more powerful than SQL Compact or SQLite.