If I wish to create .Internet application that requires database benefits, however i don't want to make use of remote database, what must i use rather?

For instance, 1) web application which must store data in certain type of database 2) desktop application which require to keep data in certain type of database

You should use SQL CE or SQLite.

There's a significant large listing of embedded databases for .Internet on Wikipedia (http://en.wikipedia.org/wiki/Embedded_database). Personally I'd choose SQLite.

Use SQLite. Here's an ADO.Internet provider with EF support: System.Data.SQLite