Code
mysqldump -u user -p database_name > dumpfilename.sql
The above command should ask you for your password once you hit enter, it'll then dump the database databasename to the file dumpfilename.sql

You can then tar the file so it's compressed; since it's just text it compresses really well, so run:
Code
tar -cvzf archive.tgz dumpfilename.sql

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