Translation:The little girl plays on the big drum.
If you say it as a hobby or regular activity, you can also say "Das Mädchen spielt (die) Trommel" but for a momentary activity and also together with what is played, I'd use auf: "Das Mädchen spielt ein Lied auf der Trommel" (with a further object like ein Lied, auf is mandatory).
The German phrasing is (etw.) auf einem Instrument spielen, meaning "to play (something) (on) an instrument". As you can see from the inflection of the article, it requires auf in the dative sense, here. This doesn't have anything to do with location, though - prepositions have many other abstract uses besides describing location.
To play something physically located over the drum would be über/oberhalb der Trommel spielen.
Danke quis_lib_duo ;)
you should have a look at the topic "adjective inflection". There are three different tables, depending of whether the phrase is introduced by a definite qualifier, an indefinite qualifier or no qualifier at all. http://germanforenglishspeakers.com/adjectives/adjective-declensions/
It's not that easy. The ending of the adjective is determined by case, number and gender of the following noun, as well as by the fact if there is no article, a definite article or an indefinite article or possessive in front.
The endings are not always the same as those for the articles.
See the full tables here:
So e.g. it is "ein kleines Mädchen", but "das kleine Mädchen".
As a rule of thumb: if the article already unambiguously denotes the case, the adjective doesn't have the respective ending. "In the given example "das" is unambiguous (can only be neuter), but "ein" is not (could as well be masculine).