- short form Polish birth certificate
- a valid ID (a passport will do)
If both of you are Polish citizens and never have been married before, then that's it. Things get more complicated if one doesn't have Polish citizenship. In that case foreigner is also required to provide a certificate of eligibility for marriage (note, that this is not same thing as certificate of being single). This certificate can usually be obtained via the consulate of the country in question. Also, if they don't speak Polish, then presence of sworn interpreter is required both when filing the papers and during the wedding.
You have to file documents with Civil Registry Office (Urząd Stanu Cywilnego) of your choosing. There is no requirement that it has to be the one with which you're registered as resident, but it sure makes the paperwork faster.
Once documents are filed and approved, you have to choose a wedding date, that is at least a month later, but no more than 6 months later (in case of civil marriage). In case of religious marriage with civil effects, you'll get certificate valid for 3 months.
For more question, check out Warsaw's Civil Registry Office FAQ (unfortunately it's only available in Polish).