Copy mysql database from remote server to local computer
I ran into the same problem. And I could not get it done with the other answers. So here is how I finally did it (yes, a beginner tutorial):
Step 1: Create a new database in your local phpmyadmin. 
Step 2: Dump the database on the remote server into a sql file (here I used Putty/SSH):
mysqldump --host="mysql5.domain.com" --user="db231231" --password="DBPASSWORD" databasename > dbdump.sql
Step 3: Download the dbdump.sql file via FTP client (should be located in the root folder)
Step 4: Move the sql file to the folder of your localhost installation, where mysql.exe is located. I am using uniform-server, this would be at C:\uniserver\core\mysql\bin\, with XAMPP it would be C:\xampp\mysql\bin
Step 5: Execute the mysql.exe as follows:
 mysql.exe -u root -pYOURPASSWORD YOURLOCALDBNAME < dbdump.sql
Step 6: Wait... depending on the file size. You can check the progress in phpmyadmin, seeing newly created tables.
Step 7: Done. Go to your local phpmyadmin to check if the database has been filled with the entire data.
Hope that helps. Good luck!
Note 1: When starting the uniformer-server you can specify a password for mysql. This is the one you have to use above for YOURPASSWORD.
Note 2: If the login does not work and you run into password problems, check your password if it contains special characters like !. If so, then you probably need to escape them \!.
Note 3: In case not all mysql data can be found in the local db after the import, it could be that there is a problem with the mysql directives of your dbdump.sql