Good question! 好问题!
First, I have to say 极了, 最, 很& 太 are all adverbs to modify adjectives.
Please keep in mind that "极了" is only used after adjectives, you can say 好极了,美极了, but you can't say 极了好,极了美。极了normally means EXTREMELY, TO ITS MAXIMUM POINT...
好极了 extremely good (it implies that something couldn't be better)
坏极了 extremely bad (it implies that something couldn't be worse)
最,很 and 太 are all used before adjectives, but in some areas people might use 很 differently, we normally say 很好,很美, but people in our Xinjiang Province might say 好的很,美的很, which sound funny.
In addition, 很 is very commonly used in daily conversation when you want to express your feeling that something is VERY good, beautiful, helpful....
"太...了" pattern is used when you want to express that something is too/ better/ worse/ less/ more than you expected. For example,
这本书很好 This book is very good. (which implies that this book might be very helpful....)
这本书太好了 This book is great. (which implies that this book is better than you expected )
It is also used to express your negative feelings that something is more than acceptable or possible:
too bad 太坏了
too noise 太吵了
too much/many 太多了
最 is used differently because it is only used in comparative sentences, in which 最 means "the most...or ..est"
最好的 the best
最快的车 the fastest car
最美的女人 the most beautiful woman
北京是世界上人口最多的城市之一 Beijing is one of the most populated city in the world.