I will be in Copenhagen from the end of August to the end of December. The seasons change so much, and I am not quite sure what kinds of things I definitely need to pack. I would really appreciate some help!
(You're the one who's studying abroad, correct?) If you don't have winter clothes already, I would just buy them in Denmark as the seasons change. Copenhagen certainly isn't a warm weather destination, but the winters there aren't Arctic blasts either. At a minimum, probably bring a few sweaters and a jacket. I haven't been to Denmark in August or September, but I have to other parts of Scandinavia and northern Germany. Late summer temperatures tend to fluctuate. Temperatures can shift from Tshirt to sweater and jacket weather and back again fairly quickly, so you probably should prepare for both. You won't need a winter coat until much later, late October at the earliest, but more likely not until mid to late November. I'm not going to comment on fashion, because I know little about what young Danes wear (other than to note that they seem to dress very well)... and because your name is somewhat androgynous, I don't know your gender...
Tom's reply is excellent. I would suggest H&M (there are three in Copenhagen) because they offer clothing for men and women, are very inexpensive, and feature actually fashionable clothes. August and September in Copenhagen are delightful. Generally in the 60s and 70s, clear, nice breeze off the water. And as noted, winters aren't dreadful. The temps reach the 50s some days, as I recall. I think average is well above freezing, at least during the day. So you don't need rough weather gear.
Maybe you've already left, but I'd definitely suggest bringing a rain jacket. And dress in light layers.