In the original language of the New Testament, the word "hypocrite" literally has the idea of an actor on stage. Back then, an actor would play multiple characters and would change roles by putting on different kinds of masks. This is an apt description of what a hypocrite looks like.
While Romans 1 deals with the "obvious" sinner- the pagan idol worshipping gentile; Romans 2 deals with a different kind of person- the religious hypocrite.
All of humanity is born into sin and actively sins. Some have just become better at hiding it than others. This is the person in Romans 2.
They may have a religious veneer or an outward "righteousness"; but in reality they are sinful by nature just like everyone else. They may not seem as bad as the person in Romans 1; but in reality they are likely in even more trouble before God because they claim to be something that they are not.
They say one thing and do another- that is the very definition of a hypocrite.
What religious hypocrites (like the Pharisees in Jesus' day) like to do is judge others. They think that by pointing out the sins of others, that they will be able to deflect attention away from their own transgressions.
They think that if they are critical of others "worse than them", that this will somehow justify them before God.
But they couldn't be more wrong.
Each of us will be held personally responsible for how we choose to live. The self righteous person on Judgment day will not be able to point at the guy ahead of him and say "what about him? He was much worse than me!"
That man will get what's coming to him, but the self righteous person will be judged as well. There is no partiality or favoritism with God.
All will get a fair trial before God.
Every word, thought, and deed will be brought forward as evidence against us on that Day. Even our secrets will be brought forward as evidence against us on that Day (Romans 2:16)!
You better make sure you find a good lawyer before that time (See 1Timothy 2:5 for a suggestion!); because it could be coming sooner than you think.