The region labeled A is Egypt.
The civilization that arose in this region was the Egyptian civilization. It was a strong long lasting civilization that left behind lots of evidence of their glorious prosper and life, most notable the pyramids.
The region labeled B is Mesopotamia.
The civilization that arose in this region were the Sumerians. The Sumerians are often seen as the civilization that set the basis of science in the ancient world, they are well known for their knowledge and inventions, most notable the development of math.