Supplication, petition, entreaty or request
Praying always with all prayer and supplication in the Spirit, and watching thereunto with all perseverance and supplication for all saints.
Ephesians 6:18 (KJV)
I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men.
1 Timothy 2:1 (KJV)
Be careful for nothing; but in every thing by prayer and supplication with thanksgiving let your requests be made known unto God.
Philippians 4:6 (KJV)
The King James Version is consistent in employing the word 'supplication' in all of the above verses. Other versions of the Bible use the words 'entreaty' or 'petition'.
In all of the versions the same Greek word deésis is being translated. The Greek word deésis comes from another Greek word deomai which means to ask or implore with passion and a heart felt desire.
The bible identifies supplication as a distinct way of seeking God. It signifies hunger and passion. This is probably why the bible asks us to talk to God with prayer and supplication. Halfhearted prayers do not get the job done.
We witness examples of petitions or supplications with Jesus (Hebrews 5:7), Zacharias (Luke 1:13) and Hannah (1 Samuel 1:9-13). For the sake of clarity. In this series on different types of prayer I will use the word supplication to refer to the passionate that the bible encourages us to exercise when praying.
Strictly speaking, supplication is not another form of prayer but rather an attitude and state of heart that should accompany our prayers.
Of course not all prayers will be accompanied with supplication. If you are asking God to prevent it from raining as you make the last few yards to your house on returning from playing football, you might not be too bothered if that prayer is not answered.
On the other hard, your supplication would be much deeper if you are asking God for a future wife or husband.
Supplication is the way we talk to God when passion and zeal are part of our prayer.