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The word ``degree'' has many meanings in mathematics. The most common meaning is the unit of Angle measure defined such that an entire rotation is 360°. This unit harks back to the Babylonians, who used a base 60 number system. 360° likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is subdivided into 60 Minutes per degree, and 60 Seconds per Minute.
See also Arc Minute, Arc Second, Degree of Freedom, Degree (Map), Degree (Polynomial), Degree (Vertex), Indegree, Local Degree, Outdegree