1. Use it to indicate degrees of arc. Don't put a space between it and the number.
2. Don't use it with temperatures measured in kelvins. This usage was abolished in 1967 by the 13th General Conference on Weights and Measures.
Write the freezing point of water as 273.15K not as 273.15°k.
3. When using it to write temperature, some organizations want you to put a space between it and the number and some don't. You be the judge.
20 °C vs. 20°C
20 °F vs. 20°F
4. 1°, 2°, and 3° are common abbreviations for primary, secondary, and tertiary in biology and medicine.
5. In medical shorthand, it is used to denote hours.
q4° means "every four hours."
Etymology - The word degree derives via Old French from the Latin de, down, and gradus, step.