For noise reduction:
Coring is used to remove fine detail information that does not contribute significantly to the detail of the picture but which adds noise to the image. Imagine the detail information viewed on a scope. About the baseline you'd see primarily the noise information, with the detail extending beyond that. Now imagine that you sliced (or cored) this signal so that only the information above the noise on the baseline came through. You would be left with most of the detail information intact but with much of the noise information removed. The coring adjustment determines how far from the baseline the detail information removed. You want to use just enough coring to reduce the noise in the picture but not so much that the fine detail in the image is affected.

For black level proccessing:
Coring is the process where pixels are evaluated against a threshold value and set to pure black if below. This is a "poor man's" noise reduction in the black areas of an image and is implemented in many digital video cameras and capture chips (like the BT848 family).