From the first chapter of the epistle to the Romans beginning in verse eighteen

First off, I would like to acknowledge that I am strolling into controversy central here because this passage is used in defence of the Church's condemnation of homosexuality. It is not my intent to directly support or dispute this save with what the Scriptures plainly tell us to be true.
With that in mind, a brief note on the preceding text. Paul had just quoted "the righteous shall live by faith" and the following verses contrast that severely.

¹⁸ For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth.

The wrath of God is not to be taken lightly, so we immediately know that the men who are declared here to be ungodly and unrighteous are in trouble. The words "ungodliness" and "unrighteousness" both mean basically the same thing. They mean anything that is contrary to God's commands. We also call this sin. We know that with this sin, they are suppressing the truth. It does not say right out which truth, but we can assume that it is the existence of God pretty reasonably based on the following verses.

¹⁹ For what can be known about God is plain to them, because God has shown it to them. ²⁰ For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse.

The Apostle Paul makes it clear that these ungodly men knew about God. He points out specifically the eternal power and divine nature (the state of being God) were clear right from the beginning. That these things were visible in the created world that we live in. Because God has been so obviously clear the whole time, the ungodly men do not have an excuse for their actions. Then Paul explains some of the results of this sin of failing to acknowledge God. 

²¹ For although they knew God, they did not honour him as God or give thanks to him, but they became futile in their thinking, and their foolish hearts were darkened.

"Futile in their thinking" would indicate that everything that they thought became totally useless. The reference to darkened hearts is metaphorical and refers to the fact that they are lost in not understanding what is right. The main point here is that the ungodly men completely lost their ability to reason and see what is right as a result of their refusal to honour God.

²² Claiming to be wise, they became fools, ²³ and exchanged the glory of the immortal God for images resembling mortal man and birds and animals and creeping things.

This is a pretty blatant reference to the practice idolatry. Idolatry is taking the true God and replacing him with something. Many modern interpretations would argue that that something can be a desire, such as sports or money. It is clear from this text that the Apostle Paul was referring to some kind of other false god. The text says that they did this as fools, as idiots despite claiming to be wise. 

²⁴ Therefore God gave them up in the lusts of their hearts to impurity, to the dishonouring of their bodies among themselves, ²⁵ because they exchanged the truth about God for a lie and worshipped and served the creature rather than the Creator, who is blessed for ever! Amen.

Paul states twice that because of this idolatry, the ungodly men have fallen into more sin. First is the impurity of the lusts of their hearts. This would indicate that they became corrupt in what they wanted. The began to desire more sin. The reference to the dishonouring of their bodies is translated right out as sexual impurity in the New International Version, whereas Young, in his Literal Translation, left in dishonouring, but also included the idea of uncleanliness. It is clear that this was a sinful act and it came as a result of them worshipping things other than God. 

²⁶ For this reason God gave them up to dishonourable passions. For their women exchanged natural relations for those that are contrary to nature; ²⁷ and the men likewise gave up natural relations with women and were consumed with passion for one another, men committing shameless acts with men and receiving in themselves the due penalty for their error.

These verses contrast what is "natural" against what is "unnatural",  more literally translated as being counter or against nature. The text identifies a natural relation for a man to be with a woman, and assumedly that means for women the natural relation would be with a man.The Apostle Paul refers to these as "dishonourable passions" when men committed these acts with other men and women likewise. Towards the end, this lust and vileness is identified as being the penalty for this idolatry. The next verses confirm that by saying:

²⁸ And since they did not see fit to acknowledge God, God gave them up to a debased mind to do what ought not to be done.

This verse reaffirms that it is the rejection of God that brought about this desire to sin. The Apostle Paul goes on to give a list of more of the types of sins that these godless men got themselves into.

²⁹ They were filled with all manner of unrighteousness, evil, covetousness, malice. They are full of envy, murder, strife, deceit, maliciousness. They are gossips, ³⁰ slanderers, haters of God, insolent, haughty, boastful, inventors of evil, disobedient to parents, ³¹ foolish, faithless, heartless, ruthless.

 A pretty intimidating list.First it says that they are filled with . . . And then again they are filled with . . . And then it characterises them by saying they are . . . The final verse of this passage offers a sort of summary to the whole point of this passage.

³² Though they know God's decree that those who practise such things deserve to die, they not only do them but give approval to those who practise them.

It states that in spite of the fact that they know what is right, they practice these things, and thus deserve to die. The final clause, though, seems to run contrary to logic. It states that they not only physically do the sin themselves, but that they also lend approval to others that sin as well. It seems that when somebody gives a "not only, but also" phrase, they intend that the latter item is worse. I would argue that the meaning of the passage is not that, but rather the passage implies that they are equally bad. It seems to say that it is just as bad to give approval to these things as it is to be physically doing them yourself.
This passage explains the issue with denying God and turning to idolatry. It gives us a clear picture of the results of this kind sin. It paints a grim picture of life without God.