how do roses reproduce?

Answer 1

They have sex ;-)

Inside the rose there are pollen sacks whose pollen is usually transported by insects to other roses and is deposited in the stigma (female part of the rose). That way seeds are produced. When they fall to the ground a new rose plant grows from it.

Look here for a detailed explanation:
http://www.scvrs.homestead.com/HybridizeKB1.html

@Freezingsymphony and Renate are also right, you could artificially grow a new rose by cutting stems and root them apart.

Answer 2

From cuttings, You take a cutting at a joint of the rose then you put it in rooting solution or powder and try to make it root

http://www.rooting-hormones.com/rose.htm

Answer 3

They reproduce by propogated cutting.