For EU nationals, the European Commission states that "the right of residence goes hand in hand with the right to work in another Member State." However, this statement does not extend to U.S. citizens, who qualify as non-EU nationals. Because the U.S. has no work agreements with the EU, for U.S. citizens, "the right to work in an EU country mainly depends on the laws of that country. " Some exceptions exist, including for non-EU nationals who have lived in an EU country for at least five years, non-EU researchers and highly skilled professionals, students and volunteers.
If you are still in America but planning your move to Europe, keep in mind that though U.S. nationals may legally stay in the EU without a visa for up to 90 days, planning for a longer or permanent stay usually involves a visit to consulate of the country where you're planning to stay. Consulates in different parts of the country have jurisdiction over their respective regions of the U.S., so you will have to visit a consulate closest to you. National laws often differ on this point--while Germany allows you to apply for a visa, or Aufenthaltserlaubnis, while in their country, France requires you to apply for a visa before you leave the U.S. Also remember that the issue of a visa may take weeks or even months.
A work permit is necessary for a non-EU national who plans to find work in the EU, and is usually distinct from the initial visa, though you may apply for them at the same time. This step is necessary even if you plan to be self-employed. Its importance is stressed by American Consulate to France: "Americans who disregard the visa requirement and apply in France for a work permit are required to leave the country to obtain the appropriate visa at a French consular office in the U.S. American citizens should not come to France expecting to find a job and to change their status after arrival. Such expectations are unlikely to be fulfilled and personal hardship may result." The process in Spain is different. To become a resident there one must first apply for a visa in America and then submit an application to authorities within Spain. As the U.S. consulate in Spain suggests, "Since regulations change continually, we suggest you check the Ministry of Interior's website, or call the Ministry of Interior, within Spain, toll-free at 060."
Americans can also gain the status of long-term resident by living in the EU for more than five years. For this privilege to be granted, you must be "in possession of a sufficient level of stable and regular resources, sickness insurance and, where required by the Member State, comply with integration conditions." The residency permit attainable after five years will allow you access to various benefits EU residents have, such as "access to employment and self-employed activity, education and vocational training, social protection and assistance, access to goods and services, etc."
Special laws exist to deal with the case of family reunification. According to the EU Commission, "legally resident foreigners can bring their spouse, under-age children and the children of their spouse." National laws may apply about the time frame for when family members may join a non-EU national who is a legal resident abroad. However, this period will not exceed two years.