Grace is a type of energy related to an angel's power and purity. It's effectively what makes them angels. Without it they are mortal.
In 4.10 Heaven and Hell, Anna Milton reveals that she is an angel who chose to fall from Heaven and ripped out her grace. Witnesses mistook Anna's grace for a shooting star. Where it landed, in Union, Kentucky, an enormous oak tree sprouted in the space of a year. Anna calls its power "pure creation." In conversation with Dean, Anna indicates that given the choice she would not recover her grace, and would remain human. She relishes the full range of emotions that humans enjoy and the choices they have.
When Anna arrives, the grace has been stolen from the oak tree, and Uriel reveals to Dean that he has it bottled in a phial he keeps around his neck. It appears as an indistinct white smoke with its own internal source of light. During the climactic fight of the episode, Anna smashes the phial, and the grace surrounds her and enters her mouth, after which her eyes glow yellow and her body glows bright white.
Anna yells for the Winchesters to close their eyes as the light intensifies. In the final flash, Alastair, and when the dust settles, Anna has also disappeared. She reappears in future episodes as a fully reconstituted angel.
Over the course of season 5, Castiel is cut off from Heaven, and by 5.02 Good God, Y'All lost a number of his angelic powers, including his ability to heal humans. He does not however appear to be any less an angel. In 5.21 Two Minutes to Midnight he has become essentially human (this is also seen with Future!Castiel in the alternate future or alternate universe in 5.04 The End, however no reference is made to his grace or lack of it. His grace is not mentioned when he is restored in 5.22 Swan Song. Following his destruction by the Leviathans he takes into himself from Purgatory, he is resurrected and lives with and marries a woman called Daphne Allen for a year, with no memory of who he is. Again it does not appear that he lost his grace.
In 8.23 Sacrifice, Metatron removes Castiel's Grace to use in the spell to expel all the angels from Heaven. He sends Castiel to Earth saying "I want you to live this new life to the fullest. Find a wife. Make babies. And when you die and your soul comes to Heaven, find me. Tell me your story."