I am not a frequent poster so cannot upload links for you here, but take a look at the on-line citizens advice/information website and also the Irish government’s Oasis website. Lots of useful free stuff there.
Mores specifically, I would recommend that if your business name is other than your own name, you need to register that name with the Companies Registration Office. Since there are two of you, I cannot see how you can avoid registering a business name. It involves a once off process for unincorporated businesses such as partnerships (cost is Euro 20 from memory) to register a business name. This will also allow you to get a relevant dot ie domain name should you wish (must easier process than applying for a discretionary dot ie domain name).
If you are going to run the business as a company (as opposed to a partnership or a sole trader with one of you employing the other), you must register the company with CRO. See its website for sole traders, partnerships and company businesses. Don't scrimp on getting advice. A good accountant used at the start of your enterprise is often the best investment you can make (disclosure - I am not an accountant, but mine have been really good and have offered practical advice including tips on bookkeeping etc and of course tax registration e.g. VAT, corporation tax and PRSI). Just remember that if you go down the company path, although you get the privilege of limited liability you also have the obligations of on-going filing requirements.