Appearances can be deceptive; things aren't always as they seem. Saul, Israel's first king, was tall and handsome, and had been anointed by God to rule over Israel (see 1 Sam. 9 & 10), but his heart didn't fully belong to the Lord. Samuel rebukes him and tells him that his rule as king will not endure. ‘The LORD has sought out a man after his own heart and appointed him ruler of his people, because you have not kept the LORD's command' (1 Sam. 13:14). This chapter is the outworking of that, although it will be several years (and 15 chapters), before Saul's reign finally comes to an end.
The first of Jesse's sons to appear before Samuel is presumably tall and handsome, and the prophet makes a superficial judgment, considering only the man's outward appearance. The Lord corrects him in v7, reminding him that ‘The Lord looks at the heart.' He rejects all of Jesse's sons, except the youngest, who although outwardly impressive (v12), has other more important qualities which were hidden from Samuel at the time. We so readily look at the outward appearance, both of people and of situations, and so often misjudge things. Paul warns us not to regard anyone from ‘a worldly point of view' (2 Cor. 5:16).
To anoint with oil was to consecrate, to mark something as holy and set apart for God. Priests were anointed, the altar and the tent of meeting were anointed, Saul had been anointed, and now David is anointed, following which, ‘the Spirit of the Lord came powerfully upon David'. ‘Messiah' means anointed, as does ‘Christ'. These are not merely extra names for Jesus, but titles packed with significance: Jesus is the Messiah, the Christ, chosen and anointed by God, full of God's Spirit, and set apart as our King and Deliverer.
The mention of ‘an evil spirit from the Lord' tormenting Saul is not easy. However, this is probably not demon possession, but something that came upon Saul from time to time; and when it did, it was not driven out by exorcism, but relieved by music (v15, 23). It seems that it was perhaps a feeling of fear and anxiety (maybe some form of depression, or what we might call a panic attack?) It may have been similar in some ways to Paul's ‘thorn in the flesh' (2 Cor. 12:7), a messenger from Satan to torment him. Whatever it was, it was in some sense ‘from the Lord', no doubt to humble and discipline Saul, and also the relief came ultimately from the Lord. Thank God for skilful musicians, and for the beauty and power of music!
Note that although the focus appears to be on Saul, there's a lot in v14-23 about David: plays the lyre, brave man and a warrior, speaks well, fine-looking, the Lord is with him, Saul liked him and was pleased with him. He's obviously someone to watch!