That would depend on a number of factors, including the system which hosts your database, the database(s) available to you, the scripting languages available on the host and the nature of what you want to do.
For a specific database application, such as a web forum, a blog/journal or a shopping cart, on a Linux host running Apache (web server) and MySQL (database server) you might use PHP to generate web pages which will connect your users to a MySQL database. For example, for a web forum you might use phpBB (http://www.phpbb.com/
), a set of PHP scripts written for this specific application. As far as the other aforementioned applications, you can find ready-made scripts for those, too. For a custom application, you'd write the PHP to generate the user interface and connect the user's input to your database.
Of course, you could use a CGI script to generate the interface and post the data to a CGI script which would handle the database transaction.
As far as a general-purpose database front end on a host with PHP and MySQL available, you might look at phpMyAdmin (http://www.phpmyadmin.net/home_page/
). Though I don't know that you intend to give users raw access to your database.
Of course, with any web interface, compatibility across web browsers would depend on the HTML you wrote or generated (as with PHP).
Without more details, I can't give you any more than that.