Nouns have different forms depending on how they are used. I think the clearest example in English is the he/she versus him/her distinction. In the sentence "I do not see this girl", the word "girl" is the object in a negated sentence (that is, it states what the subject is not doing). As a result, in Polish the noun "girl" is in the genitive form ("dziewczynki"). Since "this" is associated with the girl, it too is placed in the genitive ("tej"). If the sentence were instead "The girl does not see me", "girl" would be the subject, requiring the nominative form ("dziewczynka"), and "this" would follow suit ("ta").
kjsoda gives an excellent summary. Basically, we categorize words into serving different functions in a sentence. In English, we usually put them in the same order (subject, verb, object). That is, "the man rides the horse" means something different than "the horse rides the man". In the first sentence, the man is the subject of the sentence, while in the second he is the object (and vice versa for the horse).
In some languages, a word is modified depending on which function it is serving. You can see this in English with the personal pronouns.
She has a dog. ("She" is the subject of this sentence.) The dog saw her. (Her is the object of this sentence.) You see the same with him and her, I and me and who and whom. But note that these (and possibly a handful of others I'm forgetting right now) are exceptions. Notice that in the above sentences, "the dog" is in the same form in both subject and object.
In Polish, this is much more regular. Every polish noun has 7 forms depending on the function it serves.
I hope that between the answers, you have a better understanding now :)
As far as I'm concerned, you have to use the genitive form in negations, in sentences with nie, while otherwise you use the accusative. This would mean, that both "nie widzę ta dziewczynkę" and "widzę tej dziewczynki" were wrong. Only "widzę dziewczynkę" and "nie widzę tej dziewczynki" are right. If I've got a mistake, please correct me, but it should be right.
Follow up question: My observation is that sometimes the use of pronouns as objects changes the conventional sentence order; I believe I have seen SOV order in some preferred translations. Am I remembering correctly? If so, is there a good guideline for when to do so (e.g., do you always use SOV order in this context)?
The only thing that I'm able to say that generally pronouns shouldn't go at the end of the sentence, if this can be avoided.
So: "Nienawidzę jej" - no real alternative, it's ok. "Nie lubię jej" - still no alternative, as "Nie jej lubię" wouldn't make any sense. But if you start the sentence with "Ja", it turns into "Ja jej nienawidzę" and "Ja jej nie lubię". Because you CAN avoid putting the pronoun at the end, and putting it there would be like writing "HER" in capital letters, with a lot more emphasis.
'this girl that we are talking about'. Or maybe she's actually invisible.
Yes, I know that this is problematic, because what is perfectly natural in Polish (tej) may be kinda weird in English (this), but well, we have to keep to the direct translation here or otherwise it will be even more confusing.
You may answer with "that" any time you see a form of "tej". It's more of an interpretation, but it's correct.
"dziewczynki" is plural "girls", true.
But you also have to take cases into consideration. "widzieć" takes Accusative. A verb that needs Accusative, takes Genitive instead when negated. And just like for most feminine nouns, Genitive singular and Nominative plural are identical. And this is Genitive singular here.