1 Timothy 2:1; Communicating with God

"I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men;" 1 Timothy 2:1

I found it interesting that Paul saw the need to clarify that we need to be doing ALL of these types of communicating with God—he didn’t just say “pray”…I looked the words up in a Bible that marks key words with definitions in the back from the original Greek or Hebrew and then tried to find examples of these different forms of communication in the Bible.
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(Supplications KJV) Entreaties

 Merriam Webster: a serious request for something

When I look up "supplications" in the KJV versions, all of the verses that appear with this word have the ring of desperation in them. Job, David, Daniel and  repentant, sorry Israel all offer supplications to God, seriously asking for help.
Psalm 28:2
Hear the voice of my supplications, when I cry unto thee, when I lift up my hands toward thy holy oracle. 
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Prayers

 Merriam Webster: words spoken to God especially in order to give thanks or to ask for something

The immediate example that comes to mind is how Jesus taught us to pray in Matthew 6:
^"9 After this manner therefore pray ye: Our Father which art in heaven, Hallowed be thy name.¹⁰ Thy kingdom come, Thy will be done in earth, as it is in heaven.¹¹ Give us this day our daily bread. ¹² And forgive us our debts, as we forgive our debtors.¹³ And lead us not into temptation, but deliver us from evil: For thine is the kingdom, and the power, and the glory, for ever. Amen."
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(Intercessions KJV) Petitions 1783

 Merriam Webster: formal : a prayer or request to God or to a very powerful person or group

In the KJV it is the word intercession, and that word always has someone pleading the case of another person or group of people before the powerful one:

1. Jeremiah 27:18
   But if they be prophets, and if the word of the LORD be with them, let them now make intercession to the LORD of hosts, that the vessels which are left in the house of the LORD, and in the house of the king of Judah, and at Jerusalem, go not to Babylon.
   
2. Jeremiah 36:25
   Nevertheless Elnathan and Delaiah and Gemariah had made intercession to the king that he would not burn the roll: but he would not hear them.

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Thanksgiving
Merriam Webster: a prayer that expresses thanks to God

I think a LOT of the Psalms fits into this category!
Psalm 26:7
That I may publish with the voice of thanksgiving, and tell of all thy wondrous works.
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So I see these as meaning we should ask God for what we need (supplication,) acknowledge Him, His will, and greater plan in daily communication (prayer,) go to spiritual warfare on behalf of those who are lost or fighting around us (intercession,) and express our gratitude and appreciation to God in everything (thanksgiving!)