Linebeck helps Link save the World of the Ocean King from Bellum. He provides transport for Link in his ship, the S. S. Linebeck, and is greedy and self-centered. For the first half of the game, his goal is to obtain the Ghost Ship's treasure. After the Ghost Ship's treasure is revealed as merely a rumor devised by Bellum to attract people to the Ghost Ship, Oshus, revealed as the Ocean King, promises Linebeck one wish if he continues to sail Link across the oceans, to be granted after Bellum is defeated.
In the endgame, Linebeck saves Link and Tetra from Bellum by stabbing him, showing a rare burst of courage. Bellum, enraged, then possesses Linebeck. After Bellum is defeated by Link once again, he finally dies, and Linebeck is freed. Linebeck has been considerably humbled at this point, however, and when the now-restored Ocean King asks him what his wish will be, Linebeck first questions whether he even has to make a wish, then decides that he merely wants to have his ship back (it was destroyed by Bellum just before he possessed Linebeck), rather than asking for some great treasure.
After Link and Tetra are returned to their world, Link looks off into the distance, where Linebeck's ship can be seen sailing away.
In Spirit Tracks, Linebeck's grandson, Linebeck III, runs Linebeck Trading at the Trading Outpost in New Hyrule. Linebeck III hires a bridgemaker to build a bridge for Link, in exchange for a precious ring Link retrieves from the resting place of the original Linebeck.
Source: Wikipedia, "Characters in The Legend of Zelda series," available under the CC-BY-SA License.