Here is a pretty good tutorial on how to backup a MySQL database to move it to a different location/host:
http://www.web-console.org/upload_download_mysql_db_to_from_web_server/[
^]
If you are using Microsoft SQL, this link will help:
http://forums.asp.net/t/1493121.aspx/1[
^]
To connect to it, you will need to get the connection string from your hosting provider. Usually they provide that on the main page of the database. If not, you can always contact them for help or search their knowledgebase. Search for "Connection string" and "database" and you should find the information you need.