c. 1200, "a step, a stair," also "a position in a hierarchy," and "a stage of progress, a single movement toward an end," from Old French degré (12c.) "a step (of a stair), pace, degree (of relationship), academic degree; rank, status, position," which is said to be from Vulgar Latin *degradus "a step," from Latin de- "down" (see de-) + gradus "a step; a step climbed;" figuratively "a step toward something, a degree of something rising by stages" (from PIE root *ghredh- "to walk, go").
A word of wide use in Middle English; in 14c. it also meant "way, manner; condition, state, standing." Most extended senses in Middle English are from the notion of a hierarchy of steps. Genealogical sense of "a certain remove in the line of blood" is from mid-14c.; educational sense of "an academic rank conferred by diploma" is from late 14c. By degrees "gradually, by stages" is from late 14c.
Other transferred senses are from the notion of "one of a number of subdivisions of something extended in space or time," hence "intensive quality, measure, extent." The meaning "1/360th of a circle" is from late 14c. (The division of the circle into 360 degrees was known in Babylon and Egypt; the number is perhaps from the daily motion of the sun through the zodiac in the course of a year.) From 1540s as "a measure of heat;" the specific use as a unit of temperature on a thermometer is by 1727. In reference to crime, by 1670s as "one of certain distinctions of culpability;" in U.S. use by 1821 as "one of the phases of the same kind of crime."