People in the congregation were getting into arguments about genealogies, meanings of words, and technicalities about the law of Moses. Paul called these arguments ungodliness. These arguers probably felt that they were more righteous than anyone else, but Paul calls it ungodliness. Why? Because of what it was producing: disorder and disunity. If it was producing disorder and disunity within the congregation, it was not producing an environment for growth. Peace had gone by the wayside.
Paul also says in verse 16 to "shun" those things. Shunning denotes walking away from someone or something, turning one's back on it. These kinds of arguments do not produce the right kinds of fruit. To the contrary, Romans 14:17 says the Kingdom of God is represented by "righteousness and peace and joy in the Holy Spirit."
This is an example of ungodliness within a congregation, but a more common application appears in a context like Romans 1:18, 28-32:
For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who suppress the truth in unrighteousness. . . . And even as they did not like to retain God in their knowledge, God gave them over to a debased mind, to do those things which are not fitting; being filled with all unrighteousness, sexual immorality, wickedness, covetousness, maliciousness; full of envy, murder, strife, deceit, evil-mindedness; they are whisperers, backbiters, haters of God, violent, proud, boasters, inventors of evil things, disobedient to parents, undiscerning, untrustworthy, unloving, unforgiving, unmerciful; who, knowing the righteous judgment of God, that those who practice such things are deserving of death, not only do the same but also approve of those who practice them.
Most of these behaviors and attitudes listed here as ungodliness are what we are more familiar with. They can also occur within a congregation of the church of God. In this context, though, we find the unholy marriage of idolatry with immorality—true ungodliness.
In the context of Titus 2:11-13, we can understand that when grace truly comes into a person's life, the Christian will consciously repudiate and utterly reject ungodliness. It will not happen all at once, but if he or she is growing, the Christian will be moving precisely in that direction.