If you are already living in Poland and you intend to register yourself as a resident, the first thing to do is to go to the consulate of the embassy of your country (HERE) and get the “Certificate of registration of residence of the citizens of the European Union”. For more information about how to get it, visit the Certificate of Residence.
After you have the corresponding paper, you must go with your passport and the owner of your flat to the offices of the city council where you reside (“Urząd Miasto” in Polish) and complete the corresponding papers which they will provide you there. These offices you can find easily, you will have at least one in each district.
For me it is a little bit strange that the owner of your apartment has to go with you, but I think they do like this because you are not Polish and they want to check / verify that you are living in his/her flat. In some cases also it is OK if you have a written consent, but everything depends on the “Pani” (lady) who attends you.
When you have completed all these bureaucratic procedures, in the office they will have to give you a small paper containing all your personal data confirming that you are already registered and the number of PESEL that is a unique identification number and unrepeatable granted by the Ministry of Affairs Interns from Poland with the aim of identifying the holder through a central system of registration of persons.
During your stay in Poland the PESEL number will be very useful for many things such as opening a bank account or registering in the library in your neighborhood.
Anyway if you have any question or doubts, do not hesitate to write me.