What Does Prayer Do?

Sometimes we pray and we feel energized and sense that the presence of the Lord is close by. Other times, we feel dry and cracked, parched. We feel that the Lord is distant.
He is always near, of course. (Psalm 145:168). How we feel about it or what we sense doesn't matter much and doesn't alter the fact that He is always near. (Psalm 34:18, Matthew 28:20).
However sometimes these feelings do tend to color and tinge our communion with Him. We're human. That means we're sinful and we have an inclination to follow our deceitful heart with its emotions, rather than in trust and knowledge of the faith in God and His promises.

What does prayer do, actually? Whether we feel Him near or whether we don't, prayer is prayer and it avails much when uttered from a righteous man. (James 5:16).
1. Prayer combats discouragement.

The conclusion is clear: therefore confess your sins to each other and pray for each other. A mutual concern for one another is the way to combat discouragement and downfall. The cure is in personal confession and prayerful concern. Source: The Bible Knowledge Commentary: An Exposition of the Scriptures

2. Prayer gives strength both spiritual and sometimes physical.

In times of affliction Christians are to pray to God for help and strength. In times of blessing believers are to praise God instead of congratulating themselves (5:13b). In instances of critical sickness the sick person was to summon the leaders of the church for prayer. Prayer for the sick could result in either physical healing or spiritual blessing. In times of sin and struggle mutual intercession could promote spiritual victory. Elijah prayed with such force that God withheld rain from the earth for three and a half years and gave it again at his request. Source: Holman concise Bible commentary

3. Prayer gives us good gifts.
Don't shrink from this. We are told we have a Father who gives good gifts to His children. (Matthew 7:11). If we do not have good gifts it is because we do not ask. (James 4:2b).
Spurgeon spoke eloquently of Achsah, daughter of Caleb, who asked.

She was newly-married and she had an estate to go with her to her husband. She naturally wished that her husband should find in that estate all that was convenient and all that might be profitable. And looking it all over, she saw what was needed. Before you pray, know what you are needing. That man, who blunders down on his knees, with nothing in his mind, will blunder up, again, and get nothing for his pains. When this young woman goes to her father to ask for something, she knows what she is going to ask. She will not open her mouth till first her heart has
4. Prayer is part of the process the Spirit uses to transform our minds. (Romans 12:2). People can externally exhibit morality as if they'd put it on as a garment, without having their minds transformed as the mind of Christ. The new creation is not just a new soul, but a new mind so that we can think in righteousness and in truth. Prayer helps this transformation along.

What then do we do in obedience to Romans 12:2, "Be transformed in the renewal of your mind"? We join the Holy Spirit in his precious and all-important work. We pursue Christ-exalting truth and we pray for truth-embracing humility. ... And form the habit of meditating on the perfections of Christ. And in it all pray, pray, pray that the Holy Spirit will renew your mind, Piper, The Renewed Mind

5. Prayer nourishes us.

Prayer is the way that the life of God in us is nourished. Our common ideas regarding prayer are not found in the New Testament. We look upon prayer simply as a means of getting things for ourselves, but the biblical purpose of prayer is that we may get to know God Himself. Oswald Chambers, The Purpose of Prayer

6. Prayer is a love offering to Christ.

"But" someone says, "I don’t feel that I have any special things to pray about." Ah! My dear friend, I don't know who you are, or where you live, to not have any thing to pray about, for I find that every day brings either a need or trouble, and that I have every day something to ask of my God. But if we still insist that have no troubles, that we have attained to such a level of grace that we have nothing to ask for, then I ask, do we love Christ so much that we have no need to pray that we may love him more? Spurgeon, True Prayer-True Power

Of course there are many, many more ways that prayer works in us, in our lives, and as a method of communion with God. What are some ways you can think of?
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Further resources:
The Hidden Life of Prayer, free online book by David MacIntyre (1859-1938)
The Hidden Life of Prayer PLUS study guide and 8-week free course
4-part Sermon series, John MacArthur, Elements of True Prayer
Valley of Vision, excerpt from
The Prayer of Love
Grant me more and more
  to prize the privilege of prayer,
  to come to thee as a sin-soiled sinner,
  to find pardon in thee,
  to converse with thee;
  to know thee in prayer as
        the path in which my feet tread,
        the latch upon the door of my lips,
        the light that shines through my eyes,
        the music of my ears,
        the marrow of my understanding,
        the strength of my will,
        the power of my affection,
        the sweetness of my memory.