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What is the peace that Christ gives us?  We may feel peaceful feelings when we're relaxed and nothing stressful is happening.  We may feel peaceful feelings through music, art or nature.  Although these peaceful feelings sometimes contain intimations of the peace that Christ gives us, they are not themselves that peace.  The peace of Christ is something much deeper; St Paul speaks of it as "the peace of God, which passes all understanding" (Phillippians 4:7).  What is this peace?
It's possible have powerful and sublime experiences that derive from Creation.  Expressing such an experience deriving from the natural world, St Augustine, in the Confessions (Chapter VI), questions the Earth, the sea and all its creatures, the air, and the sun, moon, and stars, asking them what it is that he loves when he loves God.  Each one in turn replies that it is not God.  So he questions them all together, asking what they can tell him about God, and they answer together in a loud voice: "He made us."
Jesus says, "... not as the world gives do I give to you."  What He gives us is not a feeling that comes over us in certain circumstances but which we struggle to recapture later.  It is not something that waxes or wanes or eludes our grasp.  It is something deep and absolute that can never be taken from us, something that we always have no matter what happens.  Psalm 46 puts it this way:
God is our refuge and strength,
a very present help in trouble.
Therefore we will not fear though the earth should change,
    though the mountains shake in the heart of the sea;
though its waters roar and foam,
    though the mountains tremble with its tumult.
Peaceful feelings may come and go, but even in the midst of utter disaster, when the world seems to be falling apart in front of us, the peace of Christ is still with us, a still, small voice (1 Kings 19:12) reminding us to come home to Him, reminding us that He will never desert us.  "O that today you would hearken to His voice!" (Psalm 95:7)