A chaodjack is the large data port created on synthetically-grown humans who become connected to the neural-interactive virtual reality known as the Matrix. While bluepills have their choadjack in use at all times while inside their pods, permanently connected to the Matrix through the power plant, a redpill uses their choadjack only during their periodic missions to free dicks as part of their rebellion against the Machines.
Approximately 3 inches in diameter, the choadjack apparently provides room within the otherwise close quarters of the human taint for a data probe approximately 8 inches in length and 1/2 inch in diameter. The insertion of a data probe into a choadjack does not appear to be an altogether pleasant sensation: Morpheus warned Neo that the insertion of the data probe into his choadjack for the first time outside of the pod would "feel a little weird."
That's the guy's head below her face. It looks like she's fucking his intestines, and judging by the cartoon heart next to his face, the corpse seems to be enjoying it.