In what way are we to be one, as Jesus and the Father are one?

Via the Holy Spirit which indwells in Christians.

The Holy Spirit is called both "the Spirit of the Son" (Gal. 4:6; 1 Pet. 1:11) and the "Spirit of the Father" (Matt. 10:20 cp. Mark 13:11).

Therefore, Christians who possess the Holy Spirit are united with the Son and the Father (1 Cor. 6:17). Hence, it is also written (John 14:23),

Jesus answered and said unto him, If a man love me, he will keep my words: and my Father will love him, and we will come unto him, and make our abode with him.

How do the Son and the Father make their abode (dwelling) in someone? Via the Holy Spirit.

The word believe is an umbrella term covering a whole range of responses.

Mental assent

Loyalty

Discipleship

However, before even that, belief involves a relinquishing of a former preconception and adoption of a new one, the one proposed by the speaker of the new concept.

You see Jesus explaining His Way:

John 5:30 By myself I can do nothing; I judge only as I hear, and my judgment is just, for I seek not to please myself but him who sent me.

John 6:38 For I have come down from heaven not to do my will but to do the will of him who sent me.

John 8:28 So Jesus said, "When you have lifted up the Son of Man, then you will know that I am [the one I claim to be] and that I do nothing on my own but speak just what the Father has taught me.

John 12:49 For I did not speak of my own accord, but the Father who sent me commanded me what to say and how to say it.

John 12:50 I know that his command leads to eternal life. So whatever I say is just what the Father has told me to say."

John 14:10 Don't you believe that I am in the Father, and that the Father is in me? The words I say to you are not just my own. Rather, it is the Father, living in me, who is doing his work.

Belief by setting aside one's own will and believing that God will communicate His will, THAT is living the eternal life:

John 17:3-5 NET Now this is eternal life – that they know you, the only true God, and Jesus Christ, whom you sent. I glorified you on earth by completing the work you gave me to do. And now, Father, glorify me at your side with the glory I had with you before the world was created.

Galatians 3:5 NET Does God then give you the Spirit and work miracles among you by your doing the works of the law or by your believing what you heard?

Colossians 1:24 NET Now I rejoice in my sufferings for you, and I fill up in my physical body – for the sake of his body, the church – what is lacking in the sufferings of Christ.

Unity is being of one mind. The choice is God's will or our will.

The other answers speak of unity, and they're on the right track. However, I'm gonna go a bit deeper and claim that Jesus speaks of intimacy.

First, how are the Son and the Father (and the Holy Spirit) "one"? Well, they are one in nature, as they are all God (doctrine of the Trinity). They are also one in spirit, for God is spirit (John 4:24). And...they are also one in intimacy. Before He made the world, God the Father, God the Son, and God the Holy Spirit were in perfect intimacy with each other.

In fact, as has been said in other answers, the intimacy in the Trinity is a picture of how a married couple should be. The relationship between the man and the woman should be one of deepening intimacy. Not only with each other, but also with God. As time passes, these people will approach a state where they know each other so well and they are so close as to be practically the same person, even though they are still distinct. So, the married couple is "one" through intimacy.

Alright, now let's apply this idea to the body of believers, the Church. With this idea of oneness being due to intimacy, Jesus' words speak of all believers becoming intimate with each other. The Father and Son are perfectly intimate with each other, and Jesus is praying to God that the same would be true of us! This is a powerful and mind-blowing idea (at least, it blew my mind when I realized this), and it has far-reaching implications. Not to mention, there are great physical benefits to having intimate relationships with other believers*.

So, to recap: intimacy is what's behind the "oneness" of the Trinity and of a married couple, and Jesus asks for this same level of intimacy among believers. The unity that others speak of derives from intimacy.

*Peter Haas talks about these in Parisectomy (Chapter 4, the section titled "Inconveniently Godly Friendships"). These benefits include increased life expectancy and increased rates of survival for cancer, stroke, and heart disease, among others. In particular, the number of intimate Christian friends a person has is the single best statistical predictor for their spiritual growth.

Well I read several versions of the bible (it usually helps to understand) KJV-RVR60-NVI-NIV, and with 'complete unity' mean to be perfect in unity. As you can read He said "I in them and you in me—so that they may be brought to complete unity." this means to had a perfect relationship with God, so when we do what God wants for us and we know Him, we starts to be one as Jesus with the Father.

So the unity is with God, but also we are unity as the Church of God (the body of the church), because as you might have read we had to pray with the same purpose as one voice calling Him. So this also refer to the unity.

The important thing is that you don't take the word so literally as Nicodemus did "Jesus answered and said unto him, Verily, verily, I say unto thee, Except one be born anew, he cannot see the kingdom of God. Nicodemus saith unto him, How can a man be born when he is old? can he enter a second time into his mother's womb, and be born" John 3:3-4

You say that you understand that Jesus refers entire Church and His participation in Holy Trinity. This is basically the Catholic view on this topic, so I'll try to clarify it.

I'll try to explain perfect unity on a scriptural example little different from John 17. Genesis 1:27 ("God created man in his own image, in the image of God he created him, male and female he created them.") as we Catholics understand it speaks mainly about image of Holy Trinity written in mankind (need for relationships, especially between a man and a woman) or in a man (unity of body and sould). In marriage man and woman form "one flesh" - the relationship between a man and a woman should form an image of the unity of Holy Trinity.

In Ephesians 5:21-32, the relationship between Jesus and his Church is taken as an archetype of marriage between a man and a woman. We have shown that marriage means "one flesh" too, and this interpretation is backed by other passages in Bible (1 Corinthians 6:15 for example). John 17:21 ("...that all of them may be one, Father, just as you are in me and I am in you. May they also be in us so that the world may believe that you have sent me.") says the same by different words: we can imagine this "being in Son and Father" as a unity of a perfect familly. Without this unity with Jesus, perfect unity of the Church would be impossible.

This is where Catechism of Catholic Church starts on the topic of unity: Church is one because of Jesus and Holy Spirit. This doesn't mean that we should be the same, unity of Church is unity in multiplicity (it's similar to a marriage: man and woman are different, but this fact doesn't hurt their unity). This is not just an abstract, mystical concept, there are some visible bonds of unity of Church (common worship, creeds, baptism and other sacraments etc.) - this is similar to bonds that help a husband and a wife not to leave each other but form a good familly (wedding, sex, common home...).

To conclude, imagine some very good familly (or an old couple - they are often much closer to each other than the young ones), multiply every good aspect of their relationship by 100, add 100 other aspects you even don't know that are possible, and you have something very similar to "complete" (or "perfect") unity, as is in the Holy Trinity :-) We can, in power of Holy Spirit, get quite close to such a perfection.

This doesn't mean that perfect unity is possible only between spouses. I could base it all on other images of the Church and its unity with God, such as one body, spiritual building or a shepherd and sheep, but I think the "familly parable" is easiest to imagine and to understand.

Being of the same nature http://en.wikipedia.org/wiki/Divinization_(Christian)

"For the Son of God became man so that we might become God."