An eye cup is an extended piece of rubber extending from each binocular eyepiece. With some models these fold down if you wish to wear glasses. However , this has its flaws. Continuous use can damage to cups, leave finger prints on the eyepiece and even risk damage to the binoculars if incorrectly handled.
With a twist eye cup, as the title suggests, you simply rotate the eye cup clockwise or counter clockwise to extend to the required distance. This is much more simple to operate. Some are smooth throughout the adjustment, while others have increments of 2 or more to help get each one to be an equal distance from the eyepiece.
Binoculars with twist eye cups often come at a higher price than fold down versions.