There is an easier way to sum up the "Double Irish".
Ireland taxes companies based on where they are managed, but the US taxes companies based on where they realize profits and loses (as do most countries). A US company will set up an Irish subsidiary but manage it from the US, or anywhere else outside of Ireland, then transfer it's intellectual property to the subsidiary, who licenses use of said IP back to the parent company. The parent company realizes no profits in the US after paying licensing fees to the Irish subsidiary, so the US collects no taxes. The subsidiary is managed from a foreign company, so Ireland collects no taxes. That's it in a nutshell.
There are further complications where a second Irish subsidiary will be formed plus a Bermuda based shell company, but those are just for dotting the i's and crossing the t's. A further trick can be used with a Dutch company, aka. a "Dutch Sandwich", to minimize taxes even more.