It's not a matter of finding "errors". They evaluate rumors and misinformation on a variety of topics and post fact-based information that either supports or refutes. Sometimes things are true, and sometimes they are false. Just depends on the rumor.
A lot of what is being spread around about Obama is untrue, or at the very least has very little basis in fact. Even in cases where it might be mostly true, parts of it have generally been blown so far out of proportion, that it no longer resembles the facts it was based on. So it's always good to do a bit of research, and FactCheck is just one of the of the places to do that.
Factcheck.org is mostly about rumors and opinions from various people. The fact that there are posts on the site about the president does not necessarily mean that they are all facts. However, some of the posts on the site may be true to some point.