This phrase has come up a lot in the business media. It is a tax avoidance procedure that some multinational corporations use to lower their corporate tax liability. The strategy uses payments between related institutions in a corporate structure to shift income from a higher-tax country to a lower-tax country. The most popular countries being the Cayman Islands, Bermuda, Jersey and Guernsey as they have corporate tax rates of 0% – see list of Central Government Corporate Tax Rates.
Many US technology companies have taken advantage of a loop hole in Irish corporate law which allowed them to be registered in Ireland without being tax-resident there – see chart below to see how it all works. Google for instance keeps it intellectual property in an Irish company that is tax-resident in Bermuda, which has a zero tax rate of corporate tax. However from January 2015 new companies domiciled in Ireland will also have to be tax-residents there, making the Double Irish impossible.