For Mac and Windows:

Go to the website:www.mysql.com/downloads
Download MySQL community server (the RDBMS itself) and MySQL Workbench (a graphical interface) :
Choose your operating system and its version (for the mac's OS version : clic on the Apple and then This Mac) 
For Mac : Dowload the DMG archieve package ; all Apple computers built since 2007 are 64 bit . 

Run the 2 package installers :
Mysql-xxx.pkg
MySQLStartupItem.pkg
Drug the MySQL.prefPane to your /Library/PreferencePanes
Edit your .bash_profile to add /usr/local/mysql/bin by taping in your terminal : export PATH="$PATH=:/usr/local/mysql/bin"
Restart your computer
Install your MySQL Workbench by drugging it to the application folder
For more details read the ReadMe.txt !

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For Linux:

In Synaptic :
Install mysqlcommon and mysqlserver

When it asks for a password, enter one that can be used for all the users of your computer, or click enter for no password.

Using MySQL under NIS/YP requires a mysql user account to be added on the local system with:
adduser --system --group --home /var/lib/mysql mysql

You should also check the permissions and ownership of the /var/lib/mysql directory:
As root:
#cd /var/lib/mysql 
#ls -la
Should give:
drwxr-xr-x   mysql    mysql
If not : #chmod a+rx mysql

Edit your .bashrc to add /usr/local/mysql/bin by taping in your terminal : export PATH="$PATH=:/usr/local/mysql/bin"


Download MySQL Workbench from  www.mysql.com/downloads
Select the correct file for download (for 32- or 64- bits, write 'uname -a' in a terminal to see whether you are on a 32 or 64-bit machine)
Click right on the downloaded .deb file and select install