Halt!! Passport!!!

You're probably wondering at this point, well, what do I possibly need to get into Sweden? Obviously, one needs a passport to travel anywhere in the world, but as the world becomes increasingly inter-global, we're trending to facilitating world travel.
MAKE SURE YOU DON'T FORGET YOUR PASSPORT lol

There are of course different requirements for citizens of different countries. It will be more than likely that you are from a country that does not require a Visa or Permit to enter Sweden, but just in case here is a list of countries that indeed DO REQUIRE A VISA.
If you are not on this list then this means that you are allowed under Swedish Migration Laws to enter the country without any Visa, but can only reside in Sweden for less than 3 months.

So we are assuming that you were not on the list on the link provided above. Now there are three options.

Are from a Nordic Country: This encompasses Finland, Sweden, Norway, Denmark, and Iceland also referred to as Scandinavia. Citizens from these countries need no visas or residence permits and are freely allowed to reside.

EU Citizens: Citizens that are part of the EU are allowed to reside in Sweden without the acquisition of any such permits. However, if planning on staying/studying for more then 3 months then one needs to register with the Swedish Migration Board. Register here

Any non EU/EEA Citizen: Such as myself. My guess is you're from the US, well there are no exceptions for you. There are varying rules slightly, of which I am not well informed, but here are the main points.
Falling under this category, you will need to apply for a Residence Permit if your intending on residing longer than 3 months. **It will only be valid for 1 year, but can be extended**
You will need to register an account with the Swedish Migration Board and apply online here.
You can also print out an application, but this is the easiest and quickest way given you have all supporting documentation in electronic form. 

List of Supporting Documents:

• Copy of your passport
• Letter of acceptance or certificate (proof) of university
• Certificate (proof) of funds to support yourself. ** You can just send a PDF of your bank statement which can acquire through your account online. NOTE: You must show that you can support yourself by having 7,300 SEK per month (Swedish Kronas) for at least 10 months or around $1,100 per month. ---Also, if you have been given any scholarships in support for the study abroad, these would be a good supplement to provide proof of sufficient funds                  More info on supplemental documentation and visa requirements here
• Certificate (proof) of comprehensive health insurance if you are studying in Sweden for less than one year. Can just send them scanned copy of your Insurance Card.

Here are supporting links that have all the information you would need
Swedish Migration Board Online Application Form
Studying in Sweden Help
Embassy of Sweden in Washington
Swedish Consulates in USA

MY ADVICE: I would apply online with all of your supporting documentation. The application costs 1000 SEK or ~$150. It should only take around 45 mins.

I was also surprised how quickly I received a response. You will of course get a confirmation e-mail upon their receipt of your application. I received my acceptance letter within a week. It's in Swedish, I couldn't understand it, and it looks kind of unofficial, but that's it so don't worry. 

Application for Residence Permit √   Next Step:

Now you'll need to get fingerprints taken and photos so that your residence permit card can be made. I live in Florida and the closest Swedish Embassy is in Washington. I was not about to fly all the way there for some fingerprints unless you so happen to have the opportunity. There is also an embassy in San Francisco and New York.

The easiest thing to do, and it is legal, is to enter Sweden without having the Residence Permit and once you get there you can go to a Migration office and get your fingerprints taken there, just make sure it is within three months of having arrived. 

**Important: If you take my advice, just make sure you travel with your Residence Permit letter of acceptance. For US citizens I don't think it is a problem, but when I was checking in at Orlando, I was at first not allowed to board, because I had no Visa and I had no documents showing when I was returning. Maybe it's that for a Mexican Citizen, you need to at least show that you do have a round trip ticket or a return flight.

I was still allowed to board, I think it was just the dumb, senile old lady that I had the unfortunate chance of servicing me, but I would still double check with a Swedish Embassy. They can also confirm that you can do as I tell here.

Washington Embassy Phone Number +1-202-467 2600

I think their calling hours are 10-12 only on Wednesday (may have changed, but they are ridiculous so find out with anticipation)