Stubborn resistance against impossible odds by members of the Deathwing is legendary. Models that are noted as stubborn in the army lists automatically pass any Morale checks they have to take, even if faced by a situation or opponent that would normally cause them to automatically fail. However, such is their stubbornness that they will keep on fighting even when discretion may prove the better part of valour and so they may not voluntarily fall back from close combat (see the optional rule in the Designers' Notes at the back of the Warhammer 40,000 rulebook). Note that stubborn models still have to test for pinning as normal.
Characters that are stubborn confer the ability onto any unit that they lead, as long as the character is part of the unit.