In English there are four relative pronouns: that, which, who; and the genitive one: whose
You probably already know when to use which.
In Danish there are three relative pronouns: som, der; and the genitive one: hvis
When the relative pronoun is subject (as in "Jeg kender kjolen, der/som er dyr" / "I know the dress, that is expensive"), one could use either "der" or "som".
This does not apply if the word before the relative pronoun is "hvad" (what), "hvem" (who) or "hvilken/hvilket/hvilke" (which); then it must be "der".
If the relative pronoun is object (as in "Jeg kender kjolen, (som) du snakker om" / "I know the boy, (that) you are talking about); then it could be "som" or nothing.
The quick mnemonic rules
If it is "whose" in English, use "hvis" in Danish
If pne can drop out the "that" or the "who" in English, pne can keep it out in Danish or use "som"
"Hvad som", "hvem som" and "hvilken/hvilket/hvilke som" is bad Danish (or Norwegian); in these cases, use "der"
In all other cases, use either "der" or "som"
It is a used as subject, and it's not with "hvad", "hvem" or "hvilken/hvilket/hvilke"; therefore both "der" or "som" could be used.
If I understand this right (please someone correct me if I'm wrong), both explanation forget to tell if it's object of the main clause or of the subordinate clause.
So, in the lesson, they are talking about the object of the main clause.
That is the girl who loved me. Det er pigen som/der elsket mig.
Here, det and pigen both refer to the same person (pigen is attribute to the subject det because være is a stative verb). 'der' only works if the subject of the subordinate clause is the object in the main one.
However, if the object in the main clause becomes the object in the subordinate, you get a construction similar to that in English:
That is the girl (that/whom) I love. Det er pigen, (som) jeg elsker.
'I run so that you lose weight.' (I run for you to lose weight.) is an example of sentence you cannot use 'der' with. But I guess this might not be the best illustration. Hmm I'm getting confused too. I can't seem to find an adequate sentence where the object in the main is not referred to in the subordinate. It makes sense to always find the main object in the subordinate, because a subordinate clause by itself is an object of the main clause. If someone can help me clear my grammatical thoughts, that would be great! ;)
Oh btw, I tried to translate one of my previous sentences into Danish, I'm not quite happy with the result. Can someone correct me?
[...] er en eksempel af frase med der kan du ikke bruge 'der'. (Is 'der' referring to 'frase' here?) [...] er en eksempel af frase du kan ikke bruge 'der' med.
I guess now you are left with more questions than answers, sorry! :/
"som" is the relative pronoun for both subjects and objects, while "der" can only be used for subjects (here, the difference is only in style). For further reading check http://sproget.dk/raad-og-regler/artikler-mv/svarbase/SV00000049 (in Danish) and/or look at the comments above.
You are right, I turned the sentence around by mistake, because I wanted to explain it differently in the first place... /o\ I corrected it now... So there is actually a mistake in the Tips & Notes and yes, subject and object refers to case, not to properties of the referees.