See the Companies Registration Office: Required Steps to form a Company official website here. It’s quite straightforward to register a company in Ireland. You need to write a Constitution (formerly called ‘Memorandum and Articles of Association’), reserve a company name. The fastest way to register a company in Ireland is online, where you can register your company within five days.
In this process, you need to fill out form A1 to register with the Companies Registration Office (CRO) as follows:
Once you have registered, you will receive from the CRO a company identification number. You can then use this number to register for Corporation Tax with details here.
Here is a guide for for-profit company registration in Ireland. This is another great detailed source on company law in Ireland detailing how to form an Irish company.
Most non-profits in Ireland are registered as companies limited by guarantee. Non-profits can also incorporate as a trust or stay unincorporated. To incorporate as a company limited by guarantee, go through the same process in the required steps to form a company on the Companies Registry Office website, following the procedures for ‘Companny Limited by Guarentee’. Companies limited by guarantee cannot offer shares, instead, the members are ‘guarantors’, who agree to pay a nominal fee (usually 1 euro) to wrap up the business in the event of liquidation. Non-profit founders still need to submit a constitution to register. Click here for a comprehensive guide on laws regarding non-profits in Ireland.
If you wish to become a charity, you must register with the Charities Regulatory Authority (CRA) and obtain a charity registration number to obtain tax exemptions. (Source). Here is a good source detailing requirements for tax exemptions in Ireland.