If you will be studying in Sweden for more than three months and you are not an EU citizen, you need a residence permit. If you will be studying for less than three months, you may need a visa instead.
Applications for residence permit to study at universities and colleges in Sweden are handed in online at the Swedish Migration Agency's website here
If you are applying online and do not need a visa to enter Sweden, you do not usually have to visit the Embassy.
The fee differ depending on what kind of permit you apply for. The fees are available here.
The Swedish Migration Agency handles and decides in applications for residence permit in Sweden and the Embassy cannot influence the decision. If you have questions regarding student permits you can contact the Swedish Migration Agency.