You can only obtain a work permit for a specific internship. You therefore have to arrange an internship before applying for a work permit.
Students can obtain permission to do an internship within the scope of their studies 

• if the internship is directly related to the course of study and
• if they are enrolled at a foreign university or a foreign advanced technical college and if they have completed at least 4 semesters.

An internship can last up to 12 months. You can use the 12 months all at once for one internship or you can divide them up in order to do shorter internships throughout your studies. The statutory minimum wage applies to all internships unless an exception regulated by the MiLoG (law for the regulation of the statutory minimum wage) applies.

There are other visas besides the internship visa that you can apply for if you wish to do an internship, e.g. the Working Holiday Visa.

• Two fully completed application forms and declaration duly signed
• Valid passport (+2 copies of the ID page)
• Two recent passport photographs with clear background (not older than 6 months, 3.5 x 4.5 cm)
• If applicable: valid Korean Alien Registration Card
• Two copies of the internship agreement with the German employer, including all general information, e.g. type of work, intern’s tasks, working hours, monthly salary and agreements on social insurance
• Proof of qualifications (university diploma, training, work experience)
• Proof of accommodation and flight tickets
• Curriculum Vitae
• Original letter of invitation from the German employer, including company’s address and contact person
• If applicable: letter of consent from the Bundesagentur für Arbeit (Federal Employment Agency)

The documents need to be handed in personally and in original at the consular section of the German Embassy in Seoul. In individual cases, you may be required to provide additional documents.

• Two fully completed application forms and declaration duly signed
• Valid passport (+2 copies of the ID page)
• Two recent passport photographs with clear background (not older than 6 months, 3.5 x 4.5 cm)
• If applicable: valid Korean Alien Registration Card
• Two copies of the training agreement with the German employer, including all general information, e.g. type of work, intern’s tasks, working hours, monthly salary and agreements on social insurance
• Proof of qualifications (university diploma, training, work experience)
• Curriculum Vitae
• Original letter of invitation from the German employer, including company’s address and contact person

The documents need to be handed in personally and in original at the consular section of the German Embassy in Seoul. In individual cases, you may be required to provide additional documents.