Thank you and sorry (dziękuję, przepraszam)

Thanking

thank you

dziękuję is thank you — it is the 1st-person form of a verb, so it literally means I thank. thanks (informal)

dzięki is the short informal version, like English thanks.

The reply is you're welcome, don't mention it

nie ma za co — don't mention it, literally there is nothing for which.

Please, and here you are

please; here you are

proszę is the workhorse. It means please, but also here you are when you hand something over, and go ahead when you let someone speak. here you are; you're welcome

proszę bardzo is its emphatic form, and doubles as a second way to answer thank you.

formula	used for
dziękuję / dzięki	thanking
nie ma za co	replying to thanks
proszę	please; here you are
przepraszam	sorry; excuse me
nic nie szkodzi	replying to an apology

Apologising

sorry; excuse me

przepraszam covers both sorry and excuse me — an apology and a way to get someone's attention. The reply is that's all right, no harm done

nic nie szkodzi, no harm done.

OK

okej and all right, fine

w porządku both mean OK / all right; the second is the more Polish of the two.