Romans 12 begins "Therefore, I urge you, brothers and sisters, on account of the mercies of God, to present your bodies as living sacrifices..." Paul has just concluded a lengthy and in depth argument concerning God's righteousness toward both Jews and Gentiles as displayed in the gospel of Jesus Christ. He has argued that Jew and Gentile alike are on equal footing in the salvation that comes through Christ because God has remained faithful to his promises to Israel while also opening the covenant to Gentiles as well. Now, in view of all that Paul has said up to this point, Paul urges the Christians at Rome to offer their bodies as living sacrifices that are holy and pleasing to God. The languages Paul uses here seems to invoke an intentional parallel and contrast to Israel's practice of sacrifice. Israel, of course, offered sacrifices out of thankfulness for God's deliverance of Israel from slavery in Egpyt. Similarly, Paul is here urging the Christians in Rome to offer sacrifices out thankfulness for God's deliverance of them in Jesus Christ. However, the sacrifice called for here is not that of animals but for the Christians themselves to become living sacrifices; an intensely paradoxical image. These Christians' whole way of life is to be one continual sacrifice, constantly laying down their lives as Christ did. (This is especially significant in contrast to the idea that Christ's sacrifice ends all sacrifice. More precisely, Christ's sacrifice does not end the practice of sacrifice, it means that we become the sacrifice as the Spirit shapes us to be more Christ-like.)
Paul continues this line of thought by instructing the Roman Christians to "not be conformed to the pattern of this age but be transformed by the renewing of your mind". Many translations say "the pattern of this world" but the word here is actually "age" rather than "world" and this fact helps us to better understand the larger scheme of Paul's thought which is being expressed in these verses. The contrast Paul is making is not between heaven and earth as the term "world" might suggest. Instead, Paul is contrasting the present age in which the world is corrupted by sin, death, and everything evil to the new age inaugurated by Jesus Christ and symbolized by his resurrection in which this world will be transformed and all things will be made new so that life and righteousness reign rather than sin and death. Another way of saying it is that the issue here is not "Where?" but "When?". Paul is not so much saying that the Romans' behavior should be shaped by another world (heaven, for example) as he is saying that their behavior and even their very way of thinking should be shaped by another time; the time when God's kingdom will be fully established on earth. Paul is calling upon these Christians to live according to the hope of the new creation that he spoke about in chapter 8. Their behavior now is to be shaped by what they expect to be true in the future, the future where God's reign becomes complete in our world.
However, as the next few verses make clear, this offering of one's body as a living sacrifice and being tranformed by the renewing of one's mind is not just something all the Christians in Rome are to do individually or separately. In v.3-8, Paul says that these living sacrifices and renewed minds are to be working together in community like the different parts of a body. Each person has their own gifts and abilities which are not to be used to build themselves up but to build up the community of believers. Life lived in the new age inaugurated by God's messiah means living together in community. An enormous part of being a living sacrifice is continually putting the good of the community and others in it before one's own. When the community of faith lives in this way, it becomes an image of the new age for which it hopes.