Ok,
1: Are you sure MySQL is running on your local box?
2: Are you sure you have a root login? (Most likely yes)
3: Are you sure you know your root login password and have it entered correctly?
4: Can you connect to mysql using the command line client that comes with it?
If you cannot connect using the command line client then the problem is MySQL and you need to visit http://www.mysql.com and checkout the docs.
If you can connect to MySQL using the command line client but not SQLyog then we have a problem.
My guess (off the top of my head) Either it's not running or you don't have the root password set correctly.
=C=