Carefully Examining the Text

This and the previous psalm  are both usually called "royal psalms."

These psalms are linked by the Hebrew word for "salvation" in 20:5, 6, 6, 9; 21:1, 5.
The word "king" is used in 20:9; 21:1, 7. 
The "right hand" appears in 20:6; 21:8.
Psalm 21 begins and ends with a mention of the LORD and His strength

21:1-7 The LORD is source of all the king's blessings and strength

21:8-12 The enemies are conquered
Are the "you" and "your" words a reference to God or to the king? Whichever view we take may not be that important if we understand that God was the source of all the king's strength and HIs victories are only through Him.

21:13 The LORD is exalted and praised

This is called a royal psalm because of the emphasis on the role of the anointed- vs. 6- and the king- vs. 9. 

Key words in the Psalm:
"answer" in vs. 1, 6, 6, 9
"victories" "saves" "saving" or "save" are all from the same root word in vs. 5, 6, 6, 9
The word "name" is used in 20:1, 5, 7. 

The people had great hopes for the king- II Sam. 21:17; Lamentations 4:20

The king's responsibility is to trust God and to remember Him- vs. 7
Some trusted in horses and chariots but battles were won by God's power
Psalm 33:16-17, 20
Psalm 44:2-3
Deuteronomy 20:1-4
Joshua 24:11-12

C. S. Lewis called it the greatest poem in the book of Psalms.

19:1-6 The heavens are declaring the glory of God
If creation overwhelms us and leads us to stand in awe, how much more should we stand in awe of the Creator HImself.

19:2-4 They bear constant testimony to who God is. They speak a language with no words but they speak a language that has no limits or boudaries.

19:4-6 The glory is shown in the sun bursting forth from its chamber. 

19:7-11 God's glory in HIs special revelation of Scripture
The law of the LORD...                 is perfect..                      restoring the soul
This structure is found 6 times in vs. 7-9

19:12-14 In light of God's revelation. The author is in awe of His greatness and glory and  begs for HIs mercy.

The LORD is his rock and redeemer. 

Jesus fulfills the Psalm as He is Creator- John 1:1-3, 10; Col. 1:15-17

Jesus is the Word made flesh- John 1:1, 14

Jesus is our rock- Acts 4:11; Romans 9:33; I Peter 2:4-8

Jesus is our redeemer- Ephesians 1:7; Coloissians 1:13-14

In this royal psalm, David acknowledges that all he is and has is from God. 

18:3 He hears David's call and rescues him
18:16 He drew him out of many waters
18:17 He delivered him from the strong enemy
18:18 The LORD is his stay in time of calamity
18:28 The LORD is his lamp
18:29 By God's strength he performs heroic acts
18:30 He is a shield to those who take refuge in Him
18:31 The LORD is his rock
18:32 The LORD girds him with strength and makes him blameless
18:33 He makes his feet strong
18:34 He trains his arms for battle
18:35 God's right hand upholds David
18:36 The LORD enlarges his steps
18:39 He girds him with strength
18:43 The LORD placed him as head of the nations
18:47-48 The LORD executes vengeance and subdues peoples
18:50 The LORD shows lovingkindness 

This is viewed as a royal psalm, a psalm about the king.
David composed it after deliverance from his foes according to the heading. 
Look at how he describes his relationship with God in vs. 1-2.

18:1-3 He loves the LORD

18:4-6 The crisis from which the LORD delivered him

18:7-15 The LORD's dramatic intervention to rescue His servant and judge His foes

18:16-19 The LORD drew David out of the waters

Psalm 17

17:1-2  David begs God to hear his prayer

17:3-5  The LORD has examined him and tried him

17:6-8  He begs God to protect and shield him

17:9-12  David describes the wicked and their violence to him

17:13-15.  The LORD is to bring down the wicked and he longs to see their face

Connections with Psalm 16
16:1  taking refuge in the LORD. 17:7
16:8 The mention of the LORD's right hand  17:7
16:1 Keep me. 17:8
16:8 "I will not be moved" 17:5
16:5 Portion- where is our portion

Psalm 16
Heading: A Mikhtam of David- Psalms 56-60

16:1-4 David trusts in the LORD vs. those around him who trust in other gods

16:5-6 The LORD is his portion, inheritance, cup, lot, and heritage

16:7-11 The LORD is at His right hand and He will not be shaken

16:8-11 is quoted in Acts 2:25-28, 31 by Peter and Acts 13:35-37 by Paul. Both of these speakers emphasize that these words did not have their ultimate fulfillment in David. David died and was buried and his tomb underwent decay. These words have their ultimate fulfillment in Jesus and His resurrection. 

15:1 The Question: O LORD, who may abide in Your tent? Who may dwell on Your holy hill?

15:2-5a The Answer: The character of the one who continues to be in God's presence
vs. 2 Three participles describe him positively: He walks with integrity, works righteousness, speaks truth
vs. 3 negatively he does not slander with his tongue, do evil to a neighbor, or take up a reproach against a friend
vs. 4 He demonstrates who he is by the company he keeps and he keeps his word even when it is painful.
vs. 5 He does not put out his money at interest and he does not take a bribe to strike down the innocent.

15:5c The Promise: He who does these things will not be shaken.

Psalm 14:1-3 The foolishness of living without God

14:4-6 The LORD is with the righteous

14:7 O that salvation would come out of Zion.

The Hebrew word translated "there is no" as in "there is no God" is used 4 times in verses 1-3. The first time man in saying in his heart that there is no God to call him to account. The other 3 times this Hebrew word is used in vs. 1-3 is God's assessment of man: there is none good. The word is used twice in vs. 1 and twice in vs. 3. 

Psalm 14 is virtually repeated in Psalm 53.  The small differences highlight a slightly different emphasis in these two psalms. Psalm 14  has unique phrases like the LORD is with the righteous generation  and the afflicted have made the LORD his refuge.  These unique phrases are emphasized in 14:5-6. In Psalm 53 verse 5 is unique as it speaks of scattering the bones of the wicked and it stresses God's judgment of the wicked.

Romans 3:10-12 quotes Psalm 14:1-3 and Psalm 53:1-3 and emphasizes that all have sinned and are in desperate need of His grace. 

Psalm 14 ends with longing that God would bring salvation out of Zion. This salvation is through His son Jesus Christ. 

Psalm 13:1-2 How long, O LORD??
This psalm is a classic example of a psalm of lament. 
"How long" is asked 4 times with the questions surrounding it intensifying. 

13:3-4 Rescue me now

13:5-6 I will praise the LORD
The psalms of individual lament usually end in praise and the praise at the end of the psalm answers the plea expressed earlier in the Psalm.