Access is widely used as it comes as part of MS Office and Windows is shipped with an ODBC Access driver that you can use even if you don't have the Access application! One advantage of Access is that you don't need to run a server, the database file can be read from and written to by the driver. However this is not resilient for a multi-user environment and you should use MySQL, MS SQL, Oracle etc on the web.

If you do use Access on your web site then you set up the read/write permissions for the folder where it is stored. Your host should be able to provide you with information on this.