Tuition in Sweden is currently free, even for international students, but this is planned to change in 2011. Sweden plans to introduce fees for students without EU/EEA citizenship. The exact amount isn't set yet, but the rumors are that it will be around $10K/year USD. If you start your studies before the fees start, you can complete your education free.
In addition, you have to show you can pay living costs, which are much higher than in the US. The amount you must show you have in the bank is 7.300 SEK per month, or for a year's study it's 73.000 SEK (they count it for 10 months). At current exchange rates, that's about $10,500 per year.
http://migrationsverket.se/english.jsp?e…
If you plan to study undergraduate (Bachelor's), you usually need to prove your language skills in Swedish first. There is a test called TISUS you have to pass, that shows you speak/write/understand Swedish on par with a native speaker. A couple of programs are in English, but these are limited and very competitive to get entry to.
http://www.su.se/pub/jsp/polopoly.jsp?a=…
Also, Americans need a year of higher education before they can get into Swedish university.
If you already have a Bachelor's degree, it's much easier to study. There are many Master's programs available in English, which don't require Swedish first.
Foreign students are allowed to work, but this doesn't mean you will find a job (especially in Stockholm). Your Swedish needs to be very good, and most of the time you need experience or some kind of training first. The economy isn't good right now, so it's even more difficult than when I lived in Stockholm.