Hachimaki are Japanese headbands made of cloth (usually cotton), usually red or white, and often with a slogan like try hard or a picture of the rising sun. They are worn as a symbol of great exertion. These can be seen often since they are used for a wide variety of reasons mothers giving birth wear them, students studying for difficult exams, carpenters building temples, etc. Japanese competitive eater Takeru Kobayashi wore one while at a hotdog eating contest.
Hachimaki are thought to help the wearer gather their spirits and show determination/resolve. It is thought by some that the slight pressure around the forehead helps improve speed of thought and concentration. Several theories as to their origins. One theory is that they were used to ward off evil another theory is that they were just worn to keep sweat out of the eyes of Samurai.
Akane can be seen putting one on a few times in the Ranma manga in preparation for a challenge for example before her first date with Ryoga which she had originally thought as a challenge from Ukyo. The picture is taken from volume 14 of Dr Slump (very good manga by the creator of Dragonball but better in my opinion).