Celsius is, or relates to, the Celsius temperature scale (previously known as the centigrade scale). The degree Celsius (symbol: °C) can refer to a specific temperature on the Celsius scale as well as serve as unit increment to indicate a temperature interval (a difference between two temperatures or an uncertainty). "Celsius" is named after the Swedish astronomer Anders Celsius (1701 – 1744), who developed a similar temperature scale two years before his death. 
Until 1954, 0 °C on the Celsius scale was defined as the melting point of ice and 100 °C was defined as the boiling point of water under a pressure of one standard atmosphere; this close equivalency is taught in schools today. However, the unit "degree Celsius" and the Celsius scale are currently, by international agreement, defined by two different points: absolute zero, and the triple point of specially prepared water. This definition also precisely relates the Celsius scale to the Kelvin scale, which is the SI base unit of temperature (symbol: K). Absolute zero—the temperature at which nothing could be colder and no heat energy remains in a substance—is defined as being precisely 0 K and −273.15 °C. The triple point of water is defined as being precisely 273.16 K and 0.01 °C.
This definition fixes the magnitude of both the degree Celsius and the unit kelvin as being precisely 1 part in 273.16 parts the difference between absolute zero and the triple point of water. Thus, it sets the magnitude of one degree Celsius and the kelvin to be exactly equivalent. Additionally, it establishes the difference between the two scales' null points as being precisely 273.15 degrees Celsius (−273.15 °C = 0 K and 0.01 °C = 273.16 K).
Some key temperatures relating the Celsius scale to other temperature scales are shown in the table below.
(precisely, by definition)
(approximate)

Formatting
The degree Celsius is a special name for the kelvin for use in expressing Celsius temperatures.

Temperatures and intervals
In science (especially) and in engineering, the Celsius scale and the kelvin are often used simultaneously in the same article (e.g. "…its measured value was 0.01023 °C with an uncertainty of 70 µK…"). This practice is permissible because 1) the degree Celsius is a special name for the kelvin for use in expressing Celsius temperatures, and 2) the magnitude of the degree Celsius is precisely equal to that of the kelvin. Notwithstanding the official endorsement provided by decision #3 of Resolution 3 of the 13th CGPM, which stated "a temperature interval may also be expressed in degrees Celsius," the practice of simultaneously using both "°C" and "K" remains widespread throughout the scientific world as the use of SI prefixed forms of the degree Celsius (such as "µ°C" or "millidegrees Celsius") to express a temperature interval has not been well-adopted.
This practice should be avoided for literature directed to lower-level technical fields and in non-technical articles intended for the general public where both the kelvin and its symbol, K, are not well recognized and could be confusing.

Why technical articles use a mix of Kelvin and Celsius scales
One effect of defining the Celsius scale at the triple point of Vienna Standard Mean Ocean Water (273.16 kelvins and 0.01 °C), and at absolute zero (zero kelvins and −273.15 °C), is that neither the melting nor the boiling point of water under one standard atmosphere (1013.25 mbar) are longer defining points for the Celsius scale. In 1948 when the 9th General Conference on Weights and Measures (CGPM) in Resolution 3 first considered using the triple point of water as a defining point, the triple point was so close to being 0.01 °C greater than water's known melting point, it was simply defined as precisely 0.01 °C. However, current measurements show that the triple and melting points of VSMOW are actually very slightly (<0.001 °C) greater than 0.01 °C apart. Thus, the actual melting point of ice is very slightly (less than a thousandth of a degree) below 0 °C. Also, defining water's triple point at 273.16 K precisely defined the magnitude of each 1 °C increment in terms of the absolute thermodynamic temperature scale (referencing absolute zero). Now decoupled from the actual boiling point of water, the value "100 °C" is hotter than 0 °C — in absolute terms — by a factor of precisely (approximately 36.61% thermodynamically hotter). When adhering strictly to the two-point definition for calibration, the boiling point of VSMOW under one standard atmosphere of pressure is actually 373.1339 K (99.9839 °C). When calibrated to ITS-90 (a calibration standard comprising many definition points and commonly used for high-precision instrumentation), the boiling point of VSMOW is slightly less, about 99.974 °C.
This boiling–point difference of 16.1 millikelvins (thousandths of a degree Celsius) between the Celsius scale's original definition and the current one (based on absolute zero and the triple point) has little practical meaning in real life because water's boiling point is extremely sensitive to variations in barometric pressure. For example, an altitude change of only 28 cm (11 inches) causes water's boiling point to change by one millikelvin.

The melting and boiling points of water
Throughout the world, except in the U.S., the Celsius temperature scale is used for practically all purposes. The only exceptions are some specialist fields (e.g., low-temperature physics, astrophysics, light temperature in photography) where the closely related Kelvin scale dominates instead. Even in the U.S., almost the entire scientific world and most engineering fields, especially high-tech ones, use the Celsius scale. The general U.S. population, however, remains more accustomed to the Fahrenheit scale, which is therefore the only scale that most U.S. broadcasters use in weather forecasts. This has caused some confusion in the weather of Canada. When some Americans hear 32°, they think Fahrenheit, which leads to the assumption that Canada's weather is freezing at best. The Fahrenheit scale is also commonly used in the U.S. for body temperatures. The United Kingdom has almost exclusively used the Celsius scale since the 1970s, with the notable exception that some broadcasters and publications still quote Fahrenheit air temperatures occasionally in weather forecasts, for the benefit of generations born before about 1950, and air-temperature thermometers sold still show both scales for the same reason.

World-wide adoption
Unicode includes a special "°C" character at U+2103 (decimal value 8451). One types ℃ (or ℃) when encoding this special character in a Web page. Its appearance is similar to the one synthesized by individually typing its two components (°) and (C). To better see the difference between the two, below in brown text is the degree Celsius character followed immediately by the two-component version:
℃°C
When viewed on computers that properly support and map Unicode, the above line may be similar to the line below (size may vary):

Depending on the operating system, web browser, and the default font, the "C" in the Unicode character may be narrower and slightly taller than a plain uppercase C; precisely the opposite may be true on other platforms. However, there will usually be a discernible difference between the two.

See also
1) All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine. 2) Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius's name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name centigrade could continue to be used and would simultaneously inherit an intuitive association with the new name. 3) The new name eliminated the ambiguity of the term "centigrade," freeing it to refer exclusively to the French-language name for the unit of angular measurement.