In almost all cases your organization does not need to register in advance.
Each company establishes their own guidelines around the types of organizations that they'll match to. The broad categories generally include one or all of the following:
Typically most organizations don't need to register with a company. Rather an employee at a company would make a donation to a nonprofit of his/her choosing and then submit a match request to the company with the nonprofit's name and information on it. Then the company makes a really quick determination on whether the organization meets its guidelines based on the nonprofit's mission statement and IRS registration.