What does it mean "a dissertation submitted in partial satisfaction"? It appears in thesis, e.g. "A dissertation submitted in partial satisfaction
of the requirements for the degree Doctor of Philosophy in Mathematics".

Does it mean the thesis is OK for a Ph.D, just some reviewers have comments? or does it mean that it's not good enough for Ph.D, and although the one summits it will not continue to work on the thesis, but still decide to submit it for, i don't know, archiving?
This is a standard bit of text that means that the dissertation is submitted towards one of the requirements for the PhD degree. There are other requirements (e.g. coursework, oral defense), so the dissertation alone is not enough to earn the degree. 

It says nothing about the quality of the dissertation.