What does it mean to be an heir of God and a joint-heir with Christ?

Being a heir of God doesn't mean that whatever we inherit was inherited by God. For example, Bill Gates' children are each getting about 10 million dollars as their inheritance (source). Ignoring for now the fact that 10 million dollars is a really small portion of his wealth, it is certainly the case that Gates didn't inherit even that much from his parents. Thus, we can inherit salvation, eternal life in Heaven, and so on, even though God didn't get it first.

As for the joint-heirs part, NIV phrases it a bit differently:

Romans 8:16-17 (NIV)
16 The Spirit himself testifies with our spirit that we are God’s children. 17 Now if we are children, then we are heirs—heirs of God and co-heirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory.

We share in Christ's suffering because the world hates us; "If the world hates you, keep in mind that it hated me first." (John 15:18) However, we share in Christ's glory as implied by John 7:18 and John 17:22, among other verses.
However, this doesn't take into account the Jewish meaning of the word as Mason Wheeler pointed out.

A side note: the NLT version makes it even clearer and more direct.

Romans 8:16-17 (NLT)
16 For his Spirit joins with our spirit to affirm that we are God’s children. 17 And since we are his children, we are his heirs. In fact, together with Christ we are heirs of God’s glory. But if we are to share his glory, we must also share his suffering.

St. Thomas Aquinas teaches that Grace is formally participation in the Divine Nature (2 Peter 1:4). St. Thomas goes on to teach that the virtue of Charity that St. Paul writes about in the Letter to the Galatians is loving with God's love: that is, the supernatural virtue of Charity is partaking in the actual love of God Himself. Faith is partaking in the actual knowledge of God Himself (remember, the Image of God is understood as having an intellect and a will: the ability to have knowledge and the ability to make moral choices/love).

If you read Eastern Orthodox theologians, they will actaully criticize (somewhat truthfully, but really only in America) Catholics for ignoring theosis>. In fact, the whole Palamite debates of the High Medieval era were about this: the essence/energy theology is a result.

This is what C. S. Lewis writes in Mere Christianity:

"Yes, but what do we mean when we talk of God helping us? We mean God putting into us a bit of Himself, so to speak. He lends us a little of His reasoning powers and that is how we think: He puts a little of His love into us and that is how we love one another."

When we become co-heirs to Christ/ adopted Children of God, we are receiving everything God has, including His Divinity, but we are not becoming Divinity. We share in the Divine Nature, but we don't become the Divine Nature, for God became man so that men could become gods (St. Athanasius).

I'm not sure about Protestant groups other than some Anglicans and maybe the Methodists, but the Catholics and Orthodox are quite willing, and have, touched on this teaching, especially the Orthodox, due to the debate between St. Gregory and Baarlam.

The Latin Church wrote much on this before the nominalism of late Medieval period, which reached its peak in the Reformation. The Counter-Reformation emphasizes participation in Deity a lot more as a result (read St. Teresa of Avila and St. John of the Cross), and then it was de-emphasied during the Jansenism (think Catholic Calvinism) period. That period has slowly been dying (its spirit had survived very long in America), and the Latin Church, including in America, has started to more loudly teach it again.

Christi pax.

This is a really sticky bit of theology. In Jewish custom, "joint heirs" has a very specific meaning: heirs who receive an identical inheritance. There are only two churches I've seen who aren't completely unwilling to so much as touch this verse with a ten-foot pole, and both are well outside of mainstream Christianity.

The Mormons claim that this is evidence that man can be exalted and become as God, like Christ did. On the other hand, the Jehovah's Witnesses take a completely opposite interpretation: since man can receive exactly the same inheritance as Christ, and since man can obviously not become as God, this is evidence that Jesus was not divine either.

In Jewish thinking, being an heir speaks to adoption into a family. An adoption would make the adopted one legally eligible for an equal share in the father's estate.

Now Jesus was intended to be the firstborn among many brethren:

Romans 8:29

For those God foreknew he also predestined to be conformed to the image of his Son, that he might be the firstborn among many brothers and sisters.

And we believers are adopted into the family of God:

Romans 8:15

The Spirit you received does not make you slaves, so that you live in fear again; rather, the Spirit you received brought about your adoption to sonship. And by him we cry, “Abba, Father.”

With all the entitlement to an inheritance in the Kingdom of God that that entails. As joint-heirs with Christ, we have an equal share with him as Sons and Daughters of the living God.

Now, consider this, in the garden Adam an Eve were created in the image of God, with free access to the tree of life, they would have life forever in perfect communion with God... had they not grasped after God-likeness, seeking equality with God on their own terms.

Now, when talking of the resurrection Paul makes some startling statements:

1 Cor 15:42-49

42 So will it be with the resurrection of the dead. The body that is sown is perishable, it is raised imperishable; 43 it is sown in dishonor, it is raised in glory; it is sown in weakness, it is raised in power; 44 it is sown a natural body, it is raised a spiritual body.

If there is a natural body, there is also a spiritual body. 45 So it is written: “The first man Adam became a living being”; the last Adam, a life-giving spirit. 46 The spiritual did not come first, but the natural, and after that the spiritual. 47 The first man was of the dust of the earth; the second man is of heaven. 48 As was the earthly man, so are those who are of the earth; and as is the heavenly man, so also are those who are of heaven. 49 And just as we have borne the image of the earthly man, so shall we bear the image of the heavenly man.

So it is that we will be raised in the image of the risen Christ, incorruptible, immortal, redeemed, with a body like to what we have now, yet as different as that of the risen Lord who walked through locked doors and appeared and disappeared at will.

Interestingly, the Catholic church teaches, quite rightly I think, that we will be divinised with Christ in some mysterious way - "may we come to share in the divinity of Christ, who humbled himself to share in our humanity" (words of the mass, and direct teaching of John Paul II).

When the bible speaks of our adoption with Christ, it is speaking of the mystery that is God finally making us like him in a more wonderful and transcendent way after the resurrection than we ever dreamed possible when we grasped after it in the garden.

1 Cor 15:24

24 Then the end will come, when he hands over the kingdom to God the Father after he has destroyed all dominion, authority and power. 25 For he must reign until he has put all his enemies under his feet. 26 The last enemy to be destroyed is death.

For what else can be the case in a world where all evil, death and pain is removed but that every evil inclination of mans heart has been removed and we have perfect and eternal communion with the living God? Not one of us will ever sin again - the idea will never even enter our heart. Yet we will remain separate persons, but our nature will be so transformed as to be unable to sin. Now imagine what kind of a nature can it be which can last for all eternity forward never self-seeking, never sinning and never having a single conception of so much as an idea in discord with Almighty God.

That's what it means to be an heir of God and joint-heir with Christ.