우리 means "we" or "our"(possessive). There's also the humble form 저희.
내 is a contraction of 나의 (나+의)("I/me"+"'s"[possessive]). It means "my".
Due to Korean culture, to show respect, humility and a sense of community/nation, Koreans often use 우리 instead of 내/제 (depending on the situation). So it would literally be translated as "our" but the actual understood meaning is "my" (or "our" if you really meant it that way).