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1.
A band of soldiers or other combatants chosen to take the leading part in a military operation, such as an assault on a defended position, where the risk of casualties is high.

The term is from Dutch, roughly translated as "lost troop".
While the Donner Party was trapped in the mountains, a team of the fifteen strongest immigrants (five women, nine men, and a boy of twelve) set out on December 16, 1846, to find help, using makeshift snowshoes made by an old farmer, Franklin Graves. Later known as the "Forlorn Hope", the group consisted of:
*Luis and Salvador (19, 28), Miwok guides, murdered for food.
*Antonio (23), a teamster, died
*Patrick Dolan (35), died
*William Foster (31), survived
*Sarah Murphy-Foster (20), survived; she and William lost their toddler, Jeremiah (2.5)
*Harriet Murphy-Pike (18), survived; lost her baby, Catherine
*Lemuel Murphy (12), died despite his sisters; his mother, Levina (37), and brother, John (17), also lost
*William Eddy (28), survived; lost his wife, Eleanor (25), and both children, James and Margaret (3, 1)
*Franklin Graves (57), died; his wife, Elizabeth (46), and three youngest children, Jonathan, little Franklin, and little Elizabeth (7, 5, 1), were lost.
*Mary Graves (20), survived
*Sarah Graves-Fosdick (22), survived
*Jay Fosdick (23), died
*Amanda McCutchen (23), survived; lost her baby, Harriet
*Charles Stanton (30), died
The Forlorn Hope wandered from December 16, 1846, until January 17, when a Miwok village helped the seven survivors to the safety of Johnson's Ranch, where they called for a rescue mission.
On Christmas Day, hopelessly lost and their starvation rations gone, the idea of cannibalism was first discussed, but nobody could bear to kill anybody. As a blizzard lashed them that night, Antonio died, followed by Graves, who died in the arms of his daughters; Dolan went mad before he slipped into a coma. Those remaining butchered and ate the flesh of their dead companions, sobbing in shame as they ate; Luis and Salvador, plus William Eddy, refused to eat. Taking pains to avoid eating their dead relatives, the party trudged on, cursing the man whose shortcut had led them to this.
William Foster, crazed by hunger, suggested killing Luis and Salvador for food; Eddy unsuccessfully tried to discourage him before warning the two men, who ran as far as they could.
Sarah Fosdick, a newlywed, had just lost her father and then had to watch her husband die and then see his heart roasting on a stick.
"What to do we did not know. Some of those who had children and families wished to go back, but the two Indians said they would go on. I told them I would go too, for to go back and hear the cries of hunger from my brothers and sisters was more than I could stand. I would go as far as I could, let the consequences be what they might." -Mary Graves (1826-1891)
by Lorelili January 03, 2012
 
2.
The forlorn hope were the first company of soldiers to go through the hole blown in the walls of a fortress by the siege guns. Their survival was doubtful, and to lead one and live brought reputation.
Alright ye forlorn hope, once more unto the breach.
by wright May 15, 2005