Japanese: "soba-kou", literally "next-to ko", for "ko (threat) (right) next to (the ko fight itself)."
Chinese: 本身劫
A local ko threat is a ko threat that cannot be ignored, because ignoring it makes winning the ko useless. This kind of threat should therefore often be played before any other kind.
In this diagram, Black is playing ko for the life of his group in the corner. Black has just captured the ko with . The point a is now a local ko threat for White. If Black ignores this and connects the ko, White will play b, killing the black group. So Black must play at b himself to avoid this. White can now retake the ko.
In this example, Black is fighting a ko for the life of the group. When takes the ko, we can see that Black has a local ko threat at a (atari two White stones).
For clear-cut examples of why you sometimes delay local threats, see loss-making threats.