Who killed Goliath—David or Elnathan? Why is there a contradiction?

There is no contradiction—there are “two” Goliaths. The first one, the famous one, is the giant Philistine from Gath who was killed by David son of Jesse—later King David—with a stone hurled from a sling (I Samuel 17). The Bible describes him this way: A champion named Goliath, who was from Gath, came out of the Philistine camp. He was over nine feet tall. He had a bronze helmet on his head and wore a coat of scale armor of bronze weighing five thousand shekels; on his legs he wore bronze greaves, and a bronze javelin was slung on his back. His spear shaft was like a weaver’s rod, and its iron point weighed six hundred shekels. His shield bearer went ahead of him—I Samuel 17:4-7. In our measures today, Goliath stood about 9 feet, 9 inches tall. His coat of “scale armor of bronze” weighed about 125 pounds. The “iron point” of his spear weighed about 15 pounds. No matter—through God’s intervention, David took him out.
The “other” Goliath is found in II Samuel 21:19, where it says, In another battle with the Philistines at Gob, Elhanan son of Jaare-Oregim the Bethlehemite killed Goliath the Gittite, who had a spear with a shaft like a weaver’s rod—II Samuel 21:19. This was one of almost countless times where Israel was at war with the Philistines, and the equally countless skirmishes that occurred between both sides. This was another “Goliath,” not to be confused with David’s opponent. In the first fight, David was not much more than a youth. In the second one, David is king, ruler of all Israel, and El-hanan is one of his so-called “mighty men.” There is possibility that Goliath was a common Philistine name (it has been found in ancient Philistine inscriptions) or that “Goliath” was a title for a giant (much as “pharaoh” was a title and not the specific name of someone), and it seems that the Philistines boasted many giants—there was a race of giants in Canaan before the Israelites arrived, and their descendants may have been a part of the Philistine people.