Emmett is one of Edward's vampire siblings. Since Edward is frequently described as the "youngest" of the Cullens, we assume that Emmett looks around nineteen or twenty years old, and we know he's large and powerful. In fact, when Bella first tells Charlie that's she's dating Edward, Charlie mistakes Emmett for Edward and says, "I don't like the look of that big one. I'm sure he's a nice boy and all, but he looks too…mature for you" (17.118).
Emmett joined the Cullen family when Rosalie rescued him after a bear mauled him in an Appalachian forest. Badly wounded and near death, Emmett was carried by Rosalie to Carlisle to change into a vampire. Emmett is now romantically involved with Rosalie (they've been married several times), which explains why he doesn't get too close to Bella throughout the novel (see Rosalie's "Character Analysis" for more on that). Emmett seems to enjoy challenges, hence his interest in hunting bears and his confidence about hunting down and killing James. In the end, Emmett does actually kill James (with Jasper's help). Overall, he doesn't take anything too seriously, and enjoys teasing Bella.