It depends upon your needs. If you need a simple database then you could use Access. If you need something more "SQL" like then use SQL Express. But these require external software installations. Of course there are other databases as well including MySQL you could use.
If you need something embedded then use any # of embedded SQL products like Sqlite or equivalent.
For local development only then Visual Studio ships with a LocalDB instance already installed. You technically can use this on non-dev machines as installers provide it as an option but it is really just a stripped down version of SQL Express. But it isn't patched or anything so it would introduce security risks on client machines. It would be better to use SQL Express if you wanted to go that route.
Of course you don't necessarily need a local DB at all. If it fits your needs you could also consider using a cloud-based DB and host the DB in the cloud somewhere. It depends on what you intend to do with it.