If your example is meant to be a sentence, then it is ungrammatical and should have been:
The girl who is standing there is my friend.
It breaks down as follows:
The girl is my friend. Which girl? The girl who is standing there.
"who is standing there" is called a relative clause, signalled by the relative pronoun "who", which is the subject of the relative clause. The main verb of the relative clause is the present continuous verb "is standing". Where the standing takes place is indicated by the adverb "there".
You are right that in this particular case it is more or less equivalent to:
The girl standing there is my friend.
It can also work with more complicated cases, though the shorter version might sound odd to some people:
The boy [who is] writing on the board is my neighbour.
Note that the shorter form does not by itself necessitate present tense. For example:
The man [who was] sitting on our right stood up and left.
Also, shortening is also allowed when the main verb of the relative clause is the copulative ("is / was / will be") and the predicate complement (sometimes called "object") is a preposition or a past participle functioning as an adjective, such as:
The woman [who is] in this picture is my grandmother.
The book [that had been] on the table had disappeared!
The houses [that were] buffeted by the incredible wind have all fallen down.
Those [who will be] leaving next week will be given a farewell dinner tomorrow.
There are various additional cases where shortening is permitted, and this is not in any way an exhaustive list:
The people [whom] I met at the park today were very friendly.
This child [that is] here needs rest.