What the Bible says...
 Because of this, God gave them over to shameful lusts. Even their women exchanged natural relations for unnatural ones.  In the same way the men also abandoned natural relations with women and were inflamed with lust for one another. Men committed indecent acts with other men, and received in themselves the due penalty for their perversion.
...and how it can be interpreted
Dr. James B. DeYoung
Clearly Paul is condemning both lesbianism and male homosexuality. The verses come in a context in which Paul describes the general course of pagan Gentile behavior. Without excuse (v. 20) Gentiles suppress the truth of the knowledge of God revealed in the creation (v. 18-20), turn to idolatry (vv. 21-23), and degrade their bodies (v. 25). He goes on to say that such people are given over by God to a depraved mind and become filled with every kind of wickedness (verses 28-31). Worst of all they approve this behavior in others (v. 32).
Paul is like Jesus here in affirming that Christians should leave their lives of sin (see also 1 Corinthians 6:9-11). If people are oriented toward homosexuality, they should stop the behavior and seek to change their orientation.
Various attempts are used to water down the stern disapproval of homosexual behavior here. Some dismiss it as a diatribe against idolatrous Gentiles, whereas modern homosexuals may not be idolatrous (but Paul speaks more generally, not linking directly the two). Some say that Paul's words refer to ritual impurity (but such is not in the context); or that they do not condemn people who are homosexual "by nature"; he only condemns those who leave their natural orientation (yet Paul's words about "unnatural relations" parallel those used by non-Christians, such as Aristotle and Plato, to describe homosexuality in such terms). Paul indicates that all homosexuality is a departure from the created order of Genesis 1-2. Others say that Paul knew nothing of homosexual orientation, of those born with a homosexual inclination, nor did he know anything of (and thus does not condemn) life-long mutual homosexual commitment. Yet starting hundreds of years before Paul, Plato (see his Symposium) and others spoke of such mutuality, orientation and life-long commitment. It is inconceivable that Paul was ignorant of homosexual behavior and orientation, or that he would approve of such. Elsewhere, Paul "insists" that the Ephesian Christians "no longer live as the Gentiles do," who "have given themselves over to sensuality and indulge in every kind of impurity and continually lust for more." The Christians "did not come to know Christ that way." They were "taught to put off their former way of life" (which he describes in detail). There must "not be even a hint of sexual immorality or of any kind of impurity" since "these are improper for God's holy people." It is "shameful even to mention what the disobedient do in secret" (see Ephesians 4:17-5:16). Paul is like Jesus here in affirming that Christians should leave their lives of sin (see also 1 Corinthians 6:9-11). If people are oriented toward homosexuality, they should stop the behavior and seek to change their orientation.
Dr. David M. Carr
Here Paul is describing how God punished the Gentiles by "giving them over" to various forms of sexual immorality and "every kind of wickedness, evil, greed and depravity." As for other Jews of his day, one of Paul's prime examples of Gentile immorality is male-male sex (1:27). He also condemns "unnatural relations" by women (1:26), but the meaning of this is unclear. This may be sex between women, but recent research suggests that it more likely concerns women playing an unnaturally "aggressive" role in sex.
It is striking how contemporary Christians using Romans 1:26-27 fall into the same trap: focusing on the immorality they see in others when they should be seeking God's grace and love for themselves.
Paul's distaste for excessive female sexual dominance and male-male sex correspond with other ancient aspects of his thought, like his command that women not cut their hair or pray with their heads uncovered (1 Corinthians 11:5-6).
Those who isolate this as a proof text for condemning homosexuality, often ignore the point to which Paul is driving in this passage. As becomes clear in Romans 2, Paul is condemning someone for proclaiming the truth of God's judgment on Gentile sinners when "you do the same things" (Romans 2:2-3). This unknown judge is focusing on Gentile immorality when he should be focusing on his own forms of wickedness. Given this context, it is striking how contemporary Christians using Romans 1:26-27 fall into the same trap: focusing on the immorality they see in others when they should be seeking God's grace and love for themselves.