If you are trying to get a backup of your Wordpress database, it may be the easiest option to contact the tech support of your hosting company since they likely have a one click process they can use to provide a backup. Backups are often text files with SQL statements that build the database and insert the data.
Othewise, if you have SSH access, you can use mysqldump found here http://dev.mysql.com/doc/refman/5.6/en/mysqldump.html (assuming it is MySQL) This may not be as easy to understand at first, but may be the best option if tech support is not an option or you do not have control panel access to your site.
If you do not have SSH access and the database is MySQL, you can install the client on your your computer. You can find the client here http://www.mysql.com/products/workbench/ You will still need to contact tech support or use a control panel to allow a connection from your computer. This is not the easiest option, but gives you the most control. If you use this option, localhost must be replaced with the IP address of the server. You will also need port 3306 open at the hosting site.
If you have access to a control panel, then there should be a backup mechanism for the installed backup that will create a single text file with SQL statements that can build the database and insert the data. You will be able to put the file into a directory that you can FTP or HTTP download.
Lastly, if you know it is MySQL, you know what version, and you have access to the database files directly, then if you can stop the database, simply copy the files to your computer. To make work somewhere else, install a compatible version of MySQL, stop the server, and copy the backup files to a new directory. Edit the database location using a client, then restart the server.
I assume this is what you were looking for.