We are talking about grammatical gender here. They have very little to nothing to do with the meaning of the noun. E.g. all nouns which last syllable is -ung, -heit, -keit and -schaft are feminine. That leads to the situation that the word "Mannschaft" (team) is feminine although it only consists of the word for man and the ending. Another "rule": all nouns with -chen or -lein as last syllable are neuter. Why? That's why! There really is no logic. Do not try to find one.
In my opinion calling the different articles masculine, feminine and neuter raises more problems than it solves (and that is the case for French, Spanish, etc. as well), but is the established way to classify it.
Just think of it as 3 different kinds nouns the r-nouns, the e-nouns and the s-nouns. Correspondingly they have the definite articles der , die and das . Later that will help you when you have to inflect adjectives.
Like "ein schöner Mann", "eine schöne Frau", "ein schönes Kind"
This explains it much better than I ever could:
You must learn the gender along with the word, individually. i.e. instead of learning "water = Wasser", you memorize "water = die Wasser". There are a few 'tips', such as "If the word ends with -ig, -ling, -or, or -ismus, chances are it will be neutral gender (das)". But these are not even really rules, so in the end you must learn the gender individually for each word. Sorry! :)
"the girl" is not correct because "Frau" is an adult so it's a woman or a wife ( actually the correction they gave you is not really correct because "wife" is when you are married, so in this case it should be the woman). the girl is more often used for a child so it would be in german "Mädchen" i hope this helps you :)