1. A Domain name registered on you own name always will cost money. If the registrar gives you a free Domain name he has to get money from other resources. (Selling you hosting, email services, sending advertisement (spam)).
2. Be sure that the Domain will be registered on your name.
3. Be sure that you will get full control over the Domain this means:
3.1 Changing of all DNS records if the DNS server is included.
3.2 Selecting other DNS servers if you do not like the provided DNS servers or interface.
4. If you are not happy you can later switch to another registrar. Look for the costs of this change. Often you can leave free, some times you need to pay administration costs. Search on the internet if it is difficult to leave the registrar of your choice.
5. The maximum registration period depends on the top level Domain (.com .net .eu .nl). If possible do NOT register the Domain for the maximum period. If you switch to another registrar most times you need to pay the new registrar one extra year registration. If the Domain is registered for the maximum time you can not get extra registration time.
6. Domain registration does not need to be expensive (5 to 15 EURO or $ ). Check prices on the internet but do not forget to check other important information.