To export everything in the database, go to your hosting provider's control panel such as CPanel and find out how to do "an SQL dump". An SQL dump is a text file containing the SQL script to rebuild your database on your local computer. You can often do that in software your host may provide named phpMyAdmin.
Alternately you could use a desktop SQL tool; if you are on Windows one of my favorites is the free HeidiSQL; on Mac I used the commercial software Navicat for MySQL which I also used on Windows because of advanced features that HeidiSQL did not have. If you use a desktop tool you will probably have to enable remote access to MySQL from your IP address.
After you've imported your SQL dump, you can consider using this plugin to "fix up" all your URL references.
P.S. You'll need to use FTP to download all of your photos and other files you may have uploaded.