If you are using phpMyAdmin you can follow these instructions on how to backup and restore a wordpress database. After logging into phpMyAdmin, you need to choose the database that you want to backup.
Once you know which database it is you should see something like this image below:

Click the Export Tab (as shown in the image below) and then make sure that you tick (or untick) the boxes so they match the image here:

Looking at the image above this is how your page should appear -
Under Export it should all be highlighted in blue with SQL selected
Under Options there should be no ticks, with SQL compatibility mode set to NONE
Under Structure (the box should be ticked) and then -
Add Drop Table / View / Procedure / Function / Event
Add Auto Increment Value TICKED
Enclose table and field names with backquotes TICKED
Add into comments should be blank
Under Data -
Data should be TICKED
Complete Inserts TICKED
Use hexadecimal for blob TICKED
Save as file TICKED with no compression
Choose Go or Save (normally Go) and you will download to your local computer the database backup:

Restoring the backup to a new server
Create a new database and user name (if you are changing these details you will need to edit the wp_config file to reflect the changes)
Open phpMyAdmin and choose Import and make sure that your screen looks like the image below:

Job Done

Remember all you have done is backup the database. You will need to back up the files on the server where you wordpress is installed, and copy them across to their new location. You will also need to edit the wp_config file to show the changes in domain name, URL and so forth