Big Keeses are enemies that have appeared in Link's Awakening.
Big Keeses are the minions of the Grim Creeper, the mini-boss of Eagle's Tower. The Grim Creeper will send off the Big Keeses in a formation of six, and then one by one, they will fly towards Link. What Link needs to do is slash and defeat all six of them when they come flying at him.
If Link misses one of them, then the battle will continue. Each time he misses at least one, the Grim Creeper will set them in a different formation. Once Link has defeated all six of them, the Grim Creeper will fly off, leaving a warp portal behind.