Simple backup:

From one database, to a simple sql file.

mysqldump -u [username] -p [password] [databasename] > [backupfile.sql]

Backup specific tables

One database, several specified tables, to a single sql file.

mysqldump -u [username] -p …

Want to check your database size?

SELECT table_schema "Database",
 sum( data_length + index_length ) / 1024 / 1024 "Size (MB)",
 sum( data_free )/ 1024 / 1024 "Free (MB)"
FROM information_schema.TABLES
GROUP BY table_schema ;

This will work in MySQL 5.0.2 and up. For older versions, use:

SHOW TABLE STATUS