Pray for thanks and blessings

We should always thank God for all of our blessings.

Doctrine and Covenants 59:7
"Thou shalt thank the Lord thy God in all things."

Doctrine and Covenants 46:7
"  But ye are commanded in all things to ask of God, who giveth liberally...doing all things with prayer and thanksgiving." 

We pray for things we are grateful for.  We pray for things that we are striving to develop.  Doctrine and Covenants 4 "things of god must be sought after."  We can seek after Christlike attributes through prayer.  The prayer of faith we can have divine help to develop these traits.

Also, we can ask for priesthood blessings which is a prayer for us from someone holding the Priesthood .
James 5:14
"Is an sick among you?  let him call for the Elders of the church; and let them pray over him, anointing him with oil in the name of the Lord.  And the prayer of faith shall save the sick..."

We can pray for things we need help with.  Things we are struggling with and we can always have access to the priesthood for an added measure of prayer.