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:
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:
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.