Short for Metabolic Conditioning. It's the hip new term for high-intensity, low-rest Calisthenics. It's a similar idea to HIIT but without the anaerobic or the recovery periods. The purpose, supposedly, is that 60-120-second bursts of total body high intensity - or max intensity with only seconds of rest - ramp up your energy systems so that you can be prepared for anything. It's about going all-out almost anaerobically, busting past the phosphagen system and training your body's efficiency in glycolysis, etc.
For those who believe that you can burn fat with exercise (I do not), this format does keep your metabolism elevated for up to one or two hours after finishing. These sorts of things do not build strength. That's not what they are for. They are for energy, agility, and cardio endurance. Some powerlifters like to take the class because, despite their size and strength, they want agility, quickness, and endurance too. They have little of those.
I take a 50-minute class weekly which is basically Met Con but not labeled as such. A typical routine in that class might be a 20-second kettlebell swing followed immediately by 20-seconds of pushups followed by 30 seconds of mountain climbers, then a 5-second rest before repeating 3 times. Then a 30-second rest before going into the next triplet of calisthenics (which might be a similar pattern with 20-seconds of burpees, 30-second rower sprints, and 30-second squat and presses. The timing is everything.
I can testify that this format improves general conditioning, because when I began with it I couldn't really complete the class and stole seconds of rest time. I still steal a couple of seconds of rest time to catch my breath but I can get through the class.
More on the topic below the fold -
Book recommendation: 
An Ancient Four-Legged Whale
A Greek Salad seems too healthy to be tasty, but it is. 