In all this, I assume you are using MySQL as root.
See Nixon, Chapter 9, section on Backing up and restoring.
You should probably change directory to some place other than XAMPP's home (!), by putting a command like this in the shell:
cd c:\Users\<your user name here>\Desktop
This* is the command you'll need to back up that database.
mysqldump -u root -B --add-drop-database databaseName > databaseFile.sql
On Unix systems, capitalization matters, so I try to be consistent in case my web site is ever put onto Unix. Be sure that "databaseName" has the same case as you used inside MySQL. I added the "quotes" here because Unix required them -- DOS did not.
mysql -u root < databaseFile.sqlTo do this, back up on the old computer and restore on the other (below).
We won't be using passwords, but I can help if you mess up and give yourself one
(probably).
*mysqldump -u root -B --add-drop-database databaseName > databaseFile.sql:
-B --add-drop-database means "add code to create the database if it doesn't already exist, and delete it and recreate it if it doesn't." Or more succinctly: "Don't argue with me about whether the database is already there; handle it!"