PESEL is a number given to citizens in Poland (Polish acronym for „Universal Electronic System for Registration of the Population”). It is not obligatory to have this number but it may sometimes be useful to manage certain formalities such as opening a bank account, registering for a doctor or submitting your annual tax declaration.
PESEL is automatically issued to you when you register your temporary residence at your municipality/district office (pl. meldunek czasowy). If you cannot register (because for example the apartment owner does not give consent), but you need PESEL, you can apply for it.