Plain English Meaning
Class C is the controlled airspace around medium-sized, busier airports. To fly into it, you must first talk to ATC and get a response that includes your call sign. ATC will then watch you on radar and help keep you separated from other traffic.
Full Definition
Class C airspace is controlled airspace surrounding airports that have an operational control tower, are serviced by a radar approach control, and meet certain passenger or operations thresholds. It generally extends from the surface to 4,000 feet above the airport elevation, with an inner core typically 5 nautical miles in radius from the surface up, and an outer shelf typically 10 nautical miles in radius from 1,200 feet above the airport elevation up. Two-way radio communication with ATC must be before entering, and all receive including between and traffic.