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1) It is a physical object. A tall, narrow, pyramid-shaped piece of clay that is used to observe the temperature (actually the "heat-work") accomplished in a kiln.
2) It is also unit of measure, like feet or inches. In this usage, it is a way of describing how much "heat-work" is required for a certain firing process in a kiln.
Subsequently, you might say, "This clay fires to cone 10" (meaning a unit of measure). Or you might say, "I just bought a box of #5 cones" (meaning a physical object). They are both correct, and it can be a bit confusing. Note that cones do not represent just temperature. As indicated above, they represent a combination of time and temperature ("heat-work"). Although, for convenience sake, you will often hear cone and temperature used synonymously.
Everyone should fire with witness cones. These are the Large (standard) or Self-Supporting Cones that you put on the kiln shelves when firing a kiln. They will indicate the temperature ("heat-work") that location on that shelf reached, thus they "witness" the true temperature achieved. In a typical firing, the kilns temperature easily varies over one cone from top to bottom in heat, so you should put one Cone per shelf to get a good idea how you kiln is firing. It is a good idea (at least every one and a while) to put Cones on each side of your actual firing Cone. For example, for a Cone 5 firing, also put a Cone 4 and a Cone 6 next to your Cone 5 on the shelf. For Cone 05, also put an 04 and a 06 on the shelf. The lowest temperature Cone should be very bent, the middle somewhat bent (about 90 degrees), and the highest not bent at all.
If you have a kiln sitter, you also need Small cones to place in the kiln sitter. Cones placed in a kiln sitter are closer to the wall (and the heating elements), so they get hotter and would bend sooner than witness cones places on the kiln shelves. Accordingly, Orton adjusts them slightly to compensate for this. So, for a Cone 05 firing, you would put an 05 Small Cone in the kiln sitter.
Note that many people find that they have to put Cones in the kiln sitter that are 1 Cone hotter than desired to get the correct result on the shelf witness cones. For example, to achieve a witness Cone 06, you might need a Cone 05 (hotter) in the sitter. Or to achieve a witness Cone 5, you might need to use a Cone 6 in the sitter.