The "Big One" is a large zombie/monster that resides in the red marked areas in Last Day on Earth. These areas include the Infected Forest, Pine Wood and Limestone Spires. Regular weapons seem to have no or little affect on the Big One.
There are a few YouTube videos showing players killing the Big One with one shot. In these videos the players are using mods, so they're not accurate.
After killing the Big One you will receive the Z-marked CAC Card.
Let us know in the comments below how you killed the Big One.