Here is Vic Ceder's definition page:
After the 1/2 Circulate, everybody should be holding hands with at least one other dancer, and there will be four dancers in the center. These are considered the centers for the next action (and the others are considered the ends). Note that this is not a call where "centers stay centers and ends stay ends" throughout the whole call. Depending on the starting formation, the 1/2 Circulate might keep the centers in the center or not. For example, from facing lines or inverted lines, the original centers will still be the centers after the 1/2 Circulate. But from parallel waves, which is the most common starting formation, the 1/2 Circulate will cause the original trailers to become the centers (and the original leaders will become the ends).
Since there are so many possible starting formations, you probably don't want to try to memorize rules like "if I am in parallel waves and I start as a trailer I will do the second part as a center". Instead, just look carefully each time, after the 1/2 Circulate, to see who are the four dancers closer to the center. These are the ones who need to do what is described as the second action for "the centers", regardless of whether they were centers or ends originally, at the beginning of the call. (But eventually, after you have done the common cases enough times, you will begin to recognize who the centers are going to be, in those cases, without looking.)
The second action for the centers will be a Hinge. Depending on the starting position, this might be a Partner Hinge. Make sure that you can do this without hesitation.
After they Hinge, the centers will do 1/2 Box Circulate. This will leave them temporarily in a diamond. But the diamond itself isn't important -- what is important is that each dancer turns the correct amount. In particular, the leader turns exactly 90 degrees, while the trailer doesn't turn at all. You must do this precisely, in order to have the correct orientation to recognize the next thing you need to do.
If, after the 1/2 Box Circulate, you are standing next to one of the ends, then the next thing for you is to Cast Off 3/4. This might be a "push cast", in which case the important thing to remember is that you will be turning away from the center of the set.
If you are one of the other centers, you will be "moving up", to become the end of a line. You can think of this as a "phantom hourglass circulate" if it helps. Or not, if it doesn't. The important thing is that you move forward and outward, without turning.
In some cases there will be a "crash" -- you will find somebody else coming to the same spot, from the outside. (Or if you are the person coming from the outside you will find somebody coming to the same spot from the center.) A very common error in this situation is for one of the dancers to think something is wrong and turn around and run the other way. Resist the urge to do this! Unless you have some other reason to believe that you did one of the earlier parts incorrectly, just go there and take right hands with the other person.
The point of doing this is not to attempt to memorize all these possibilities. It is, rather, to help you get used to the idea that running into a Partner Hinge, a Partner Trade, a "push cast", or a "crash" is a perfectly normal thing. You should not consider it surprising, and you should not try to "fix" anything to avoid doing it.