A database may be accessed using the Internet and some "open web" sites may be managed as databases behind the scenes, but often the term "database" means a formal collection of records comprising...
The Internet is a giant database with many different databases used for multiple things: information, useless facts, homework help, ect. Databases are what make up the Internet
No, if you run a database on localhost, and you are the only one to edit it from your local pc... but if you want others to read/write to the database you will need an internet connection.