To throw your hat into the ring is to enter a fight or contest. The term comes from a time when men wore hats, and informal bare-knuckle boxing
matches were held in alleys and the back rooms of bars. A man interested in competing would literally throw his hat into the ring created by the spectators
to announce his intention to fight. It can also be seen as a challenge
The term became the unofficial moniker of Eddie Rickenbacker's 94th Aero Squadron. Their logo, painted on the side of all their aircraft, was a depiction of Uncle Sam's Stars-and-Stripes Top Hat sideways with the opening facing the viewer in a vertically-oriented ring.
The term has expanded today to include any contest, especially a political race.