Under cover as if he was just going to make a sacrifice, Samuel was sent by God to ordain a new king for Israel. This happened while the old king was still in power. It was risky business for the old prophet. Samuel already knew enough about Saul's ideals to fear that Saul would kill him if he found out what he was up to over in Bethlehem of Judah.
So, along with all the leaders of Bethlehem, Samuel went to the house of one, Jesse. We must assume that this occasion was incredibly significant, since long after these incidents we can find God referring to Himself as "the root and the offspring of David" (Revelation 22:16 & 5:5) and even as the "root of Jesse" (Isaiah 11:10 & Romans 15:12).
Now, Samuel was quite surprised by God's choice. God did not pick the oldest son of Jesse nor the most attractive. He picked the baby of the family, a rough young man named David. Not that David was particularly unattractive, but his appearance was not the basis of his selection. It was his heart. In I Samuel 16:7 God instructed Samuel, "Look not on his countenance... for man looks on the outward appearance, but the LORD looks on the heart."