For danbooru purposes a circle cut (サークルカット) is a special type of advertisement for a doujin circle. There are two basic types pictures tagged as circle cut. The traditional variety has some distinct features:
There is always a white box with a black border in the upper left corner.
The circle name is displayed prominently on the cut, as well as textual information about what the circle intends to offer.
They are (usually) in some standardized size, as specified by the event.
They are monochrome for the printed event catalog, however the circle cuts are usually in color for web catalogs.
The box is there for specific information, such as when the doujin circle will be attending an event and where their stand exactly is. The box is also used by event goers as a checkbox, often with a highlighter.
The second kind of picture doesn't need to conform to the visual template exactly, but is characterized by having been released as a direct preparation of the exposition in question. They might vary in appearance from just a character's portrait + circle name, up to entire small scenes or comics whose purpose is to announce the booth number. For example, see post #634255 and post #161515.
