Registration (zameldowanie) is very important in Poland. You need to register for a temporary stay, and then for a residence permit (karta pobytu), for the authorities, banks, and for the fiscal authorities.
Everybody who comes to Poland needs to register within three days of entry. You have to apply to the appropriate registration office (Urzad Meldunkowy). If you spend your time in Poland at a hotel, hostel, etc. the registration will be done for you. If you stay with friends, you will have to go to the registration office with the owner or tenant of the flat at which you are staying. Nevertheless, registration is fairly easy.
For permanent residence in Poland, the registration process is more complicated. You will need your passport or ID-card and a rental agreement. In addition (although this doesn’t appear to be a consistent requirement), you may need to be accompanied by the owner, who has to show proof of ownership (e.g. certificate of title) and his identity card.
This might seem a bit exaggerated, but with the zameldowanie come several privileges (at least for Polish people), and that´s why the process is so complicated. After you have obtained temporary registration (zameldowanie tymczasowe), you can apply for a residence permit (karta pobytu) at your local wojewoda office.