Can anyone recommend an off the shelf internet database so we can all log in around the world - up to 5 users?

Our criteria is easy to use, reliable and cost effective. We also need to be able to run report on tasks recorded each week.

Many thanks

Um. Are you wanting to share your internet connection?

If you're talking about collaborative task and/or project management & tracking, you might look at the following (some are free, some inexpensive, some quite expensive):

TaskBin (http://www.taskbin.com/todolist/default)
Hiveminder (http://hiveminder.com/splash/)
Nozbe (http://www.nozbe.com/gtd/business)
Zohoprojects (http://projects.zoho.com/jsp/home.jsp)
Basecamp (http://basecamphq.com/)

You wan to create a database or host one? To create and run one I would look at DB Tutorials - Connect, Retrieve and Display information from SQL Server and other data sources (http://dbtutorials.com) for hosting one I would check out serverintellect.com Good (http://www.serverintellect.com)luck!