K-cups are part of the new generation of coffee brewing, where you use pre-packaged "pods" to brew one cup of coffee at a time rather than using loose grounds to brew a whole pot. In this case, the K-cup is used exclusively by the Keurig coffee brewing system and can't be used with any other coffee makers. The Keurig operates differently than a drip coffee maker, and uses pressure to get the brewing done faster. That's what makes it reasonable to actually just brew your cup of coffee when you want it. You only have to wait a few seconds, so why settle for stale coffee sitting on a warming plate? This is the main selling point of a single-serve coffee system.
The K-cup is a plastic cup sealed with foil. The filter and grounds are all packed inside, so you just put the whole thing in the machine. It pierces the lid, and forces the water through. When the brewing is finished, you just take out the container and toss it. There are no wet filters or grounds at any point. Since all the grounds are contained, you can switch from one brew to another without having to stop and clean the machine. Many coffee makers can also make tea and hot chocolate as well as just coffee. You can buy most K-cups in boxes of 20 to 24, or bulk packs of 50. Some variety packs let you try several kinds without having to commit to a whole box of just one.