Brinksmanship is particular style of behavior during negotiations, especially
crisis diplomacy. It in essence is an international
game of chicken, where a nation takes actions that deliberately move the situation closer to the brink of war. This is done in the hope that the other side will re-assess the potential gains and punishments to be had, and decide that the issue in question is more important to our original nation actor and thus yield the issue.
The problem with brinksmanship, of course, is that it depends on the acting nation having correctly predicted that their escalatory action will not, in fact, move the situation into open conflict. This implies that even if they have guessed wrong about the target nation's priorities (i.e. the issue is important enough for them to go to war) either they (the actor nation) see such a conflict as favoring them, or that there will be time enough after the threat to back down again and avoid the war.
Probably the most famous example of brinksmanship is the Kennedy Administration's handling of the Cuban Missile Crisis, at least in public. The reality of the situation appears to be that there was a 'carrot' (removal of U.S. Jupiter IRBMs from Turkey) offered to the USSR as well as the publicly brandished 'stick' of nuclear attack.