EU nationals: You have to bring a European health insurance card (EHIC) with you to Dresden.
Non-EU citizens: Do not obtain a health insurance policy in your home country for the duration of your stay in Dresden. You must purchase a health insurance policy at a German public insurance agency AFTER you arrive in Dresden.
US citizens: Your public health insurance in the U.S. may be accepted by a German public insurance during your sojourn in Dresden. Please ask your insurance company in the U.S. concerning this question.

Fore more information about the Health Insurance in Germany see: Health Insurance in Germany.

For general information on the German Social Insurance System have a look on the webpage Deutsche Sozialversicherung (English).