How do Catholics reconcile Christ's intercessory prayer with the doctrine of the Trinity? John 10:30 (KJV)  is often quoted to support the doctrine of the Trinity:


  30 I and my Father are one.


But during the great intercessory prayer, Christ asks God for his disciples to also be one with them:


  John 17:20-21 (KJV)   
  
  20 Neither pray I for these alone, but for them also which shall believe on me through their word;
  21 That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.


Non-trinitarians often quote that scripture to show how John 10:30 (and other scriptures like it) is being misinterpreted. They say, "If God and Christ are literally one, then is Christ praying for us to become part of their nature too?"

Tons has been written about the Trinity, so surely a reasonable explanation for this exists. 

How can John 17:20-21 be reconciled with the doctrine of the Trinity?
The fundamental thing you have to remember about the Trinity is that God is both One and Three. It's really easy to find quotes that support Jesus and the Father as separate, and the response of the Christian is "yes, they are". It's also really easy to find verses that completely support the idea that God is One, and that Jesus is God. And the response of the Christian is "yes, they are". It is precisely the combination of those that gave rise to the doctrine of the Trinity.

A good reference for this is the Athenasian Creed:


  The Father Uncreate, the Son Uncreate, and the Holy Ghost Uncreate. The Father Incomprehensible, the Son Incomprehensible, and the Holy Ghost Incomprehensible. The Father Eternal, the Son Eternal, and the Holy Ghost Etneral and yet they are not Three Eternals but One Eternal. As also there are not Three Uncreated, nor Three Incomprehensibles, but One Uncreated, and One Uncomprehensible.