Rejoice in the Lord always; again I will say, rejoice.  Let your reasonableness be known to everyone. The Lord is at hand;  do not be anxious about anything, but in everything by prayer and supplication with thanksgiving let your requests be made known to God.  And the peace of God, which surpasses all understanding, will guard your hearts and your minds in Christ Jesus. (ESV)At a typical prayer meeting, what do we do? ... We share requests. We have request boxes. We make lists. We submit them to one another, share them through prayer chains, and ultimately lay them at the feet of God. This is precisely what we ought to do, according to this verse, but there is more here. Why did Paul use so many different words to describe our prayer request list?
Prayer: προσευχή (proseuchḗ) is literally speaking about prayer. It is the act (or the place where) we come to God to communicate. It comes from another word that means prayer, but includes the idea of nearness (the prefix pro).
Supplication: δέησις (déēsis) is, according to Thayer's Greek dictionary, need, want, privation, a seeking, asking, entreating... it is an entreaty to God or to man. When you dig into the roots of this word, you realize that it is a word that demonstrates the binding of yourself to someone because you have a need/desire and they have the means to fulfill. It is what our little ones do when we are at the store and they see that candy they want and they wrap their arms around our leg and begin to ask for that candy. (Not in the obnoxious way that some children do this, but in a humble entreaty to the parent.)
Thanksgiving: εὐχαριστία (eucharistía) is just what it sounds like. It is a giving of thanks, in the midst of the entreaty. It is acknowledging that the king has already been so good to us. It is what surrounds our entreaty, and reminds us that we have come to the right person. It is recognizing the good and attributing it to the God who gave.
Requests: αἴτημα (aítēma) is the well-worded petition. This is when we put our exact words to the desires. This is our list, our box, our specific need or want...
Put it all together... We are coming near to God to communicate with Him. It feels one way, but it is conversational. We cling to Him because He is our King and the fulfiller of all that is good. We acknowledge that He has already done this in the past and that He will continue to do this in the future. And we lay our specific desire at His feet, knowing that he will fulfill in the best way possible... and usually in a way that is better than we can imagine.
When captured like this, the ending of this passage makes sense. When coming to God in this way occurs, there is a peace that arrives with it. Don't be mistaken, it is not a peace that we have manufactured, but a peace that is given, simply because we have been granted an audience with the Good King of the Universe.