Employment as defined by the Foreigners’ Employment Act refers to a working relationship and also applies to apprenticeship situations, to lessons performed, and to other employee relationship such as made in a free service contract.
With some residence permits (permanent resident – EC or red-white-red card plus an employment may be exercised without further notice.
Information about full and part-time employment, freelance work, independent contractors, seasonal workers, apprenticeships, agency workers, and voluntary workers may be found at http://www.migration.gv.at/en/living-and-working-in-austria/working/kinds-of-employment.html
If none of the residence permits apply, then under the Aliens Employment Act, you need a work permit. A work permit is limited to a particular job and a particular activity, and is limited (maximum 1 year) issued. It is requested and obtained from the government by the prospective employer.
You may receive a work permit if you were legally employed in the past 14 months for a total of 52 weeks and you have a red-white-red card.
With an exemption certificate and if you are legally settled, you may work throughout the country. An exemption certificate is valid for five years and may be issued when during the previous eight years you worked for at a least five years in Austria, you attended the last year of school in Austria, and have a parent who was legally established in Austria for at least three out of the last five years.
For detailed information, please contact the Chamber of Labour (in German).
Austrian immigration law and policy is explained online here: http://legislationline.org/topics/subtopic/33/topic/10/country/44
Austrian laws may be explained in English here http://www.ris.bka.gv.at/defaultEn.aspx
More information on working in Austria (www.help.gv.at)