I don't understand when you use them and what the difference is between the
three.

For example, when you say:

> 火事 **によって** 多くの森林が焼けてしまった。  
>  今回の地震 **による** 津波の心配はない。
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2011-11-05T05:20:16.177
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question
grammar word-choice て-form
What's the difference between による, により and によって?
Depending on the context, they can all mean:

  * **because of/due to** (a reason):

> **震災{しんさい}によって** たくさんの人の命が失われた。  
>  There were many people who lost their lives **due to the earthquake**.

  * **by** (a method/way/means):

> 先生は **テストにより** 、[学生]{がくせい}が[理解]{りかい}したかどうかをチェックする。  
>  Teachers check if students understand **by (means of) tests**.

  * **by** (identifies the agent of the action, particularly in a passive sentence):

> この機械{きかい}は **誰によって** [発明]{はつめい}されましたか？  
>  **Who** was this machine invented **by**?  
>  
>  (just for comparison, here's that as an active sentence without `によって`):  
>  この機械を発明したのは誰ですか？  
>  Who invented this machine?

  * **dependant on** (when highlighting differences):

> 生活{せいかつ}習慣{しゅうかん}は **国によって** 違う。  
>  Lifestyles differ **depending on the country**.

`によって` and `により` are effectively the same [according to
Daijirin](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E3%81%AB%E3%82%88%E3%82%8A&dtype=0&dname=0ss&stype=0&index=115020900000&pagenum=1),
but `により` is somewhat more formal than `によって` [according to
Daijisen](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E3%81%AB%E3%82%88%E3%82%8A&dtype=0&dname=0na&stype=0&index=14147100&pagenum=1).
I'll try to break up your example sentence (and add one for `により`):

> **火事{かじ}によって** 多くの[森林]{しんりん}が焼{や}けてしまった。  
>  **Because of a fire** many forests burnt down.  
>
>
> 牛は **角{つの}により** 人は **言葉により** 捕{つか}まえられる。  
>  An ox is taken **by the horns** and a man **by word**.

`による` is slightly different (i.e. `による` is used in the following example
because it's talking about a _tsunami_ because of the earthquake):

> 今回の **地震{じしん}による津波{つなみ}** の心配{しんぱい}はない。  
>  There's no worry of a **tsunami because of the earthquake** this time.  
>
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2011-11-05T05:55:26.053
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による modifies a noun, and both により and によって modify a verb or an adjective.

により and によって are almost synonymous, although in the informal context, によって is
more usual than により.
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2011-11-06T00:42:55.810
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answer
I ended up reaching this post because I was looking for the answer while
reading through the grammar but I think I finally understand.

Like user797 pointed out, によって is depending on/according to.

But whether or not you were asking this, I think want to know the difference
from that compared to both による and によると. I'm assumign this because I was the
most confused about this. The book I'm using specifically says that AによるとB
means The information from 'B' depends on the actual source which is 'A'.

人によって意見は違う。 Opinions are different depending on the people. (no direct source)
彼によると悪い判断だった. According to him, it was a bad decision. (direct source is him)
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answer