In the Bible it talks of Jesus being one with the Father John 10:30 but also says that we should be one with them John 17:11. In the Bible's original Greek text, is the same word used for both, or is the word that describes our oneness with them different? The scriptures that speak of our heavenly Father and Jesus as being one are often used to prove the doctrine of the Trinity. (The Trinity being of one nature and one substance.) Does the original Greek word denote a physical oneness in substance, or a oneness in unity, purpose, action, and nature?
Working in unison, living in unity.
Mark 10:8 8
Judges 20:11 (ASV)
Oneness usually implies oneness of mind, actions, and so forth.
1 Corinthians 1:10 (In contrast to modern setting.)
Colossians 3:13-14 (Note relationship between love and unity (oneness))
In a deeper sense, just as a Husband and Wife are ONE through a God-ordained arrangement, the Christ is ONE with the congregation. This principle is especially important, because its necessary in saving through unity:
1st Corinthians 7:14 (Note the saving aspect of oneness)
In this way the congregation (Christ's bride, loved as his own body) is also cleansed by the Christ due to marital contract and attains everlasting life.
Conversely, the "Harlot" separating, disuniting from the Christ, unifying with Satan's world, becomes one with the world, and they share sins all the way back to Abel's murder. Luke 11:51, Revelation 17:2 COMPARE TO: 1 Corinthians 6:16
From the contents of your question I am assuming you are making reference to:
John 17:22 KJV
Very often in order to understand the meaning of any particular Scripture it is necessary to take into account many other Scriptures, and that is true in this case. This verse must be read along with not only what is said in chapter 17, but has to reach back to what the whole situation was at the time and especially what Jesus was conveying to his disciples.
In order to take in the whole concept we have to start with:
John 13:1 KJV
Beginning here Jesus is telling them not only what is going to happen, not only in the immediate future, but giving them encouragement for the persecution to come.
He begins by showing them how important humility is by washing their feet, and gives more emphasis to it when:
John 13:6, 7 and 8 KJV
So what does humility have to do with oneness?
Matthew 20:28 KJV
Here Jesus is telling them; that in order to be like him, they have to give up self and serve others.
John 13:14 KJV
He goes on further to say:
John 13:16 KJV
And again you wonder what does that have to do with oneness. well just hang on I'll get there.
John 5:19 KJV
Here Jesus is telling them that he does not arbitrarily do things, that the father is a part of everything he does, and that they are patterned after the ways of the father.
John 13:34 and 35 KJV
The purpose of this command is pretty self evident in that in order to represent Jesus they must assume the characteristics of Jesus.
John 14:10 and 11 KJV
Here Jesus is saying in effect by now you should know by now, after seeing all my miracles; that the father and I are part of the same God, but if you don't believe me you at least ought to believe it because of the miracles.
John 14:19 and 20 KJV
Here Jesus is telling them that once he goes to the Cross and ascends to the father, that he will again be spiritually united with the father, and that his spirit will be united with them, so that they all become a common spirituality.
From this point up until chapter 17 he warns them of the travail about to beset them, and so that they will not be blindsided and lose faith. And in the first part of chapter 17 he is in effect telling them that after he has gone to the cross and ascended that he will give them a new eternal life.
John 17:1 and 2 KJV
Finally we get to your Scripture:
John 17:21 and 22 KJV
So to answer your question what Jesus apparently meant when he said "be one" was he was referencing being one Spirit for eternity.