A breakout is a military operation to end a situation of encirclement or siege. It is used in contexts such as: "The British breakout attempt from Normandy".
A breakout is achieved when the encircled forces, and its allies (perhaps from outside the encirclement), attack a weak point in the encirclement, creating a breakthrough, and then move through that gap to freedom.
An example is the battle of Hube's Pocket on the Eastern Front in World War II, where the German First Panzer Army was encircled by Soviet forces, but broke out by attacking west, linking with the II SS Panzer Corps, which was breaking into the encirclement from outside.
Of necessity, the broad concept is subject to interpretation. In The Blitzkrieg Myth, John Mosier poses interesting questions as to whether the concept as applied to tank and other warfare in World War II was more misleading to planning than helpful, on account of the numerous exceptional conditions faced in war, and also whether evaluation based largely on how well breakout or breakthrough potential was realized is appropriate.