Are you allowed to work in Sweden? Work Permits explained.
Eu/EFTA citizens are allowed to live and work in Sweden without any permits but you must register your right to residence online - http://www.migrationsverket.se/English/Private-individuals/For-EU-citizens/Work/How-to-register/Register.html within 3 months of arriving in Sweden.
For non-EU citizens you must have been offered a job before you can obtain a work permit. You are not allowed to enter Sweden until the permit is granted, and you cannot come to Sweden to look for work without a permit.
To obtain a permit you must have a valid passport, a genuine offer of work at the same level as Swedish collective agreements, been offered a salary that is at least on the same level as Swedish workers and have been offered a monthly pretax salary of at least SEK 13,000.
The permit is approved based on the salary of one job and cannot be an amalgamation of several lower paid roles. Some individuals will need a visa as well as a work permit if they are a National of one of the countries in this list - http://www.government.se/content/1/c6/13/59/55/c1144799.pdf
In common with other countries, temporary workers in some professions are subject to special rules e.g. performers, au pairs, visiting researchers, athletes and seasonal workers – for more information see here - http://www.migrationsverket.se/English/Private-individuals/Visiting-Sweden.html