This article only covers backing up from the Command Linux (i.e., the shell, usually bash or korn (ksh) shells).
If you wanted to back up the database timsdata, you would simply use a command like:
mysqldump -c -u tim -p timsdata > timsdata.sql
This would create a file called timsdata.sql in the current directory. You should be aware that where I've entered the name tim in italics, you should specify another authorized user for the database.