Paul talks about how God evaluates both the Jews and Gentiles (non-Jews) fairly. God has high standards because God does not like sin screwing up others' lives (not to mention that heaven is a place specifically designed to be sin-free). Some of the readers may be familiar with the Mosaic Law that the Jews tried to follow, but the Mosaic Law can be foreign to the Gentiles. Nevertheless, God still has a case.
Paul writes that those who "sinned without law will also perish without law" and those who "sinned in the law will be judged by the law" (2:12). For those under the law, only "doers of the law will be justified" (2:13) -- think of how bank robbers who are aware of the law yet still rob the bank are not good. But for those not under the law, those people have "conscience also bearing witness", with "their thoughts accusing or else excusing them" (2:14-15). Basically, Paul is talking about the general sense of right and wrong that people have. People disagree about what is right and wrong all the time, but in the end, we have done some things and later realized that we shouldn't have done such things due to how wrong they were. In other words, we can't even follow our own standards 100% of the time -- so there's no way we can even match God's perfect standards!
The judgment happens "in the day when God will judges the secrets of men by Jesus Christ" (2:16). That's another way we know that God's evaluation is just -- we can't even hide or tamper evidence because God can just know the evidence. Based on God's judgment to sinners, the result is permadeath (because evil lasting forever is not good). But God provided us a way for us to not face permadeath (though we will die, we will get restored). Paul goes more into detail in the next chapter.