How to export database from Amazon RDS MySQL instance to local instance?

To export db from RDS

mysqldump -h rds.host.name -u remote_user_name -p remote_db > remote_db.sql

When prompted for password, provide the password

To import db on RDS

mysql -h rds.host.name -u remote_user_name -p remote_db < remote_db.sql

When prompted for password, provide the password

Sure.

Take the dump from the remote RDS Server:

mysqldump -h rds.host.name -u remote_user_name -p remote_db > dump.sql

When prompted for password, provide the password for user=remote_user_name (remote server)

Upload it to your local mySql instance:

mysql -u local_user_name -p local_db < dump.sql

Also, if you own an ec2 server in the same region, I'd suggest take a dump there. zip the file and then scp it to your local machine. Typically, the compressed version of the file would be much smaller and you'd be able to transfer it quicker.