Ginnarbrik is a very minor character in LWW; in fact he is not even named in the book. The name "Ginnarbrik" was given to him in all the film and TV adaptions.
What we do know about him was that he was the White Witch's chief servant and driver of her sleigh. He accompanies her on all her journeys and is present at all the key moments featuring the White Witch. He was with her the day she met Edmund, and travels with her and Edmund while she searches for the missing Pevensies.
He is cruel to Edmund, and restrains him while the White Witch prepares to kill him as he is now only a nuisance. The plan is thwarted, however, when a group of Narnians come to rescue Edmund. Ginnarbrik escapes unscathed when the White Witch has the presence of mind to make them both appear as tree trunks in order to escape notice.
While not implicitly stated, it's natural to assume that Ginnarbrik is present at the killing of Aslan and the subsequent battle of Beruna. He is never mentioned again, and his fate is unknown.