In Context

1 Samuel 16:7

But the LORD said unto Samuel,

Look not on his countenance, or on

the height of his stature; because

I have refused him: for the LORD 

seeth not as man seeth; for man

looketh on the outward appearance,

but the LORD looketh on the heart.

Samuel is the prophet at the time of this story. He was the same boy that Hannah prayed for and promised up to the Lord, and who answered "Here am I" when he heard His voice as a child. Saul was the king, and had seriously offended the Lord by his disobedience. So the Lord sends Samuel to anoint a new king from the sons of Jesse. (Note that his being anointed means that he would not immediately take on the role as King, but would do so at a later time.) 

Because it was dangerous to announce what was happening, Samuel is told to make his errand that of going to worship. He takes an offering, and invites Jesse and his family to join him, where he examines the sons to determine who is to be anointed. 

Samuel believes it is to be Eliab.

Then this memorable admonition is given.

Samuel then looks at each of the sons and determines through the Spirit that the Lord had not chosen any of them. He then asks if there are any sons not present, and determines that the youngest, David, is out tending the sheep. 

David is brought to him, and is anointed. 

Interestingly, in verse 12  he is described as "ruddy, and withal of a beautiful countenance, and goodly to look to."

Saul, having lost the Spirit of the Lord is afflicted with evil spirits. In the text, it says an "evil spirit from God." However, we know that God never sends evil spirits to afflict and torment. The Joseph Smith Translation clearly denotes such in verses 14, 15, 16, and 23 by inserting "which is not of God."

The servants of Saul mention that there is a son of Jesse who plays the harp, and is "cunning in playing, and a mighty valiant man, and a man of war, and prudent in matters, and is a comely person, and the LORD is with him." (verse 18)

They send for him and David becomes Saul's armour-bearer. Whenever Saul is afflicted by evil spirits, David plays the harp and "Saul was refreshed, and was well, and the evil spirit departed from him." (verse 23)