We wait for the manifestation, the removal of the veil of the fleshly nature and the redemption of our bodies.

The quote is below, but it's easier to understand when it's in plain 21st century English:

Romans 8:18-23 I, Paul, can see that the sufferings of this present time are not worthy to be compared with the glory that shall absolutely be seen in us when the veil of our flesh is removed.  For the whole hearted expectation of the creation itself is waiting for the unveiling of the matured sons of God.  For the creation and all the creatures were put into bondage to the present futility, pain, and suffering that you can observe.  Creation didn't go willingly but went because of the one who, seeing the vision of hope, put them into bondage.  The reason the creation has this great expectation is that it knows something: Creation itself will also be delivered from the bondage of corruption into the glorious liberty of the mature sons of God.  For we know that the whole creation groans and is going through birth pains right up to the present.  It's not only the creation that is going through these birth pains, though.  We are feeling the same pain, we who have the firstfruits of the Spirit.  Yes, we groan within ourselves.  We groan because we are waiting to become mature sons of God and we are waiting to be placed into ruling positions of authority, which is the same thing as saying that we are waiting for the redemption of our bodies.

Let that sink in.

Romans 8:18-23 For I reckon that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed [Apocalypse, Unveiling] in us.  For the earnest expectation of the creature waiteth for the manifestation [Apocalypse, Unveiling] of the sons [huios, mature sons] of God.  For the creature was made subject to vanity, not willingly, but by reason of him who hath subjected the same in hope,  Because the creature itself also shall be delivered from the bondage of corruption into the glorious liberty of the children of God.  For we know that the whole creation groaneth and travaileth in pain together until now.  And not only they, but ourselves also, which have the firstfruits of the Spirit, even we ourselves groan within ourselves, waiting for the adoption [Eastern adoption is the placing of a mature son into a ruling position], to wit, the redemption of our body.

Adam and Eve sinned.  Since God had set them into a ruling position over all of the creation, all of creation "was made subject to vanity" with them.  Now, we wait for the unveiling, "the redemption of our bodies."  When our bodies are redeemed, then the whole creation will be delivered, too.  We, being mature, will be set into ruling positions in the Kingdom of God.  The immature will not rule and reign with Christ.  There is a saying: sometimes maturity comes with age, and sometimes age comes all by itself.  Pray that we are faithful so that we can take part in this.