They mean basically the same thing, "to correct." You can interchange them without affecting the meaning, but there are subtle nuance differences. Of the four, 纠正is the most direct (with the harshest meaning), and it calls to mind a military officer correcting a cadet because of the word "纠". (Or a teacher correcting a pupil). 修正 is the most "gentle" way to say to correct, because it means to "edit and correct." 更正is the most formal, and you'll find that word used in the sense of "erratum" in a newspaper article. Finally, 改正 is a pretty neutral term.

纠正：指改正错误、不公正或对标准的偏离。例句：我们必须纠正错误，修改相关法规中对女性的歧视条款。（为了表示强调，这个地方不能用”改正“） 改正：适用于大多数情况。语气程度比纠正略弱。 更正：替换性质的改正，公文性质的申明比较常见。例句：原作者名李梦潮应为李孟潮，特此更正。 修正：在原有的基础上改正一部分。近义词：改正、改进、调整。例句：之前我们的方向是正确的，但我们的做法太幼稚，犯了一些错误，需要修正。

纠正: to point out where you are wrong and correct it . 改正: to correct. 更正: to alter,or modify it so that it is correct. 修正: to alter,or modify it in order to make it more appropriate,proper,suitable etc for something. 个人愚见.经供参考.

纠正 is used to correct some behavior which is not that good. E.g : ①你必须纠正你的坐姿。 but sometimes you can use 错误 which is a abstract Noun after it E.g：②纠正你的错误 改正 is more often used in oral language,when students make some mistakes in homework ,we say:改正作业中错误 更正 is more official but its meaning is close to 改正 修改 : the 3 word above are used when you do make some mistakes or something, but when 修改 appears, that does not always mean you have made mistakes, it's can used when asking you to make your essay better e.g 你的作文写得好，但是有些地方还需要修改一下，会更好

纠正通常是用于“别人纠正你的错误"的情况；而改正和更正的主体可以是你也可以是别人，而且是将错的改成了正确的；修改的主体也可以是你或别人，但修改以后的不一定是正确的，也就是说修改可能是将正确的改成了错的，也可能是将错的改成了正确的。