Is the sentence 오늘(은) 제 생일입니다 right? And if it is, what is the meaning of 제 in this sentence?
It would be more perfect if you use “오늘은 제 생일날입니다”. Here it’s added “생일+날” because it’s more emphasis on the day that the speaker want to mention. Plus, through my experience living in Korea, I many time heard Korean people say “생일날” when they want to mention the day.
Back to your question “제”. It means “my”. “제” is the polite word and advanced word in Korean language and culture. Remember that, if speaker speaks to stranger or person in higher position than him/her or the one who is older than him/her, using “제” makes the speaker look polite and knowledgeable in Korean language.
In addition. “제” is shorten from “저의”. You can use both of them exchangeably.