If you want to know how to move a whole site database from one host to another (say, from Godaddy to Dreamhost), this article is a must read. It will show and explain the whole process of exporting and importing a database (or to be more precise, all of its tables) using phpMyAdmin, the default MySQL database administrator. It is meant for those who don’t have Administrator access to phpMyAdmin (and cannot create databases from inside phpMyAdmin).
Let’s begin. First, you need to export your website’s database tables from the old host. To do so, access phpMyAdmin via your hosting panel (it’s Domain->Databases->DBWebAdmin for Plesk, and CPanel should have something similar), click on the database name (so that you see all the tables in the main window), then click on the “Export” tab at the top.

You should see the following export window. Click “Select All” to select all the tables and make sure the SQL radio button is checked. At the bottom, check “Save as file”, put in a file name (in this case it’s “mydb”), select Compression: “zipped” or “gzipped” to make the file smaller (this will help a lot if your host has a limit on the size of the database and tables you can upload) and click “Go” to get the file ( you may have to wait for a while, especially if the web server is slow and the file is big). Optionally, you may change the other parameters if your host told you that you need to or you know what you’re doing.

After you’ve saved the file, you need to create a new database at your new host. It is very desirable to assign it the same name, username and password as the old one; because otherwise your site won’t work and you’ll have to change the database settings inside the website configuration file (usually config.php or something like that). Go to your hosting panel and create it (I assume you know how to do it).
After it’s ready, open the phpMyAdmin for the new database and click on the database name ((like you did for the old one), only this time there should be no tables present and you should continue by clicking on “Import”. Here you need to select the file you just saved to your computer and click on “Go” (the other options are good by default in most cases).

After a while (depending on the speed of the server and your Internet connection), you should see an “Import has been successful” message and the tables of your database appear in the left window.

In the rare case when your database is bigger than the upload limit, you will have to export and import tables part by part (i.e. export/import the first 20, then the next 20 and so on). Kind of hard, but at least you can do it.
Now all you have to do is copy the contents of the “httpdocs” or “public_html” to you new host via FTP and optionally modify the files which contain the database name, password and username (search Google to find out how to do it for your software, like “how to change database username in Joomla”), then, if you already moved your domain name or registered another one (and modified the proper files), you should be able to access your old site on the new host. Congratulations!