A database that lives in a file on the computer itself, so the program works without internet and the data stays on that machine.
With a local database, invoices, products and customers are saved in a file on the hard drive of the computer where the program is installed. There is no server, no account and no connection: the program reads and writes directly, so it is fast and never waits for a network. The alternative is a cloud database shared by several computers, which needs internet and sign-in.
Choosing between them is a practical decision, not a technical one: is the program used on one computer or several? When you build a Windows program in Fanann you pick either a local database, which works offline on one PC and which you back up yourself, or a cloud database shared between computers. Installing the same program on two PCs with local data gives each PC its own separate data, not shared data. Practical tip: start local if you have a single counter, and find the backup and restore feature in the program before you enter your first real invoice.