理解:understand (formal). You are learning something that you dont understand. After your hard working with your brain, you 理解了 it.
e.g.
a teacher teaches his students some theory. After that the teacher asks: 你们理解了吗?
瞭解:know
Some information that you don't know. Then somehow(someone told you or you saw some messages etc) you know it.
e.g.
通过对这篇文章的阅读,他瞭解到中国是一个很大的国家。You CANT use 理解 here because reading is NOT hard working with brain.
明白:understand(informal)
the teacher can also ask like this: 你们明白了吗?
You see, teaching a theory to students is formal case, but telling someone some information is informal case ---- here we can use 明白.
e.g.
你在给Peter写信吗?不行,你现在还不能给他写信,因为(some reason),你明白了吧?
懂:equal to 理解 and 明白
懂 can be used for formal case and for informal case.
領悟:after a very long and hard time, someone 领悟到了 something.
e.g.
His father told him something when he was a child. At that time he couldn't understand what his father meant. Now he is 40 years old and he got it. ----- 他领悟到了他父亲当年说的那句话的意思了。
領會:a formal word, meaning "understand something unclear, implied, psychic".
e.g.
Your boss has a meeting with your team and he says: I'm a very nice person. I never give anyone any deadline. I just tell you guys: If you could finish this job before March, it would be great.
Now you can say: 我领会到了Boss的意思:如果我不能在三月份之前完成工作,我就会被开除。
This is not a very good example because this is not very formal. In fact you can use 我明白了Boss的意思 or 我懂了Boss的意思 too.