I've frequently encountered problems with しまう. Usually it appears alongside て
form, like 遅れてしまいました, "Unfortunately I came too late". In this case, it
signals regret on the speakers behalf.

I'm having far more issues with this verb when its like here:
母からの手紙は箱の中に大切にしまってあります。

I can't even translate this sentence. I think the speaker says that his mom's
letter is precious to him out of all the things in the box. But really, I
don't know how to incorporate a meaningful interpretation of しまう here. Maybe
like "ultimately is precious"? The て-form with ある also kind of gives me a
headache. Maybe it means that the letter has reached the status "precious" and
stays in that status now? It's one of the functions I learned for て form + いる.
Like 窓が困っている (or ある, because its an object? I once learned that いる is for
living beings and ある for objects. But I feel like いる might have been combined
with non-living objects too already. My memory might deceive me on this
though...^^).
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grammar
How does しまう function here and/or what does it mean?
I think here it's a different thing. It'a not the -てしまう in the sense of
expressing regret for something (or doing something throughout, completely).

Rather, it's the verb しまう (editing my previous answer, as Shoko pointed out
here the verb is 仕舞う) that means "to be put away, to lock/store, etc.. So

> 母からの手紙は箱の中に大切にしまってあります

could be translated as " _the letter from mom is safely closed/stored inside a
box_ ".

EDIT: I was interrupted while writing and I forgot to add that -てある simply
indicates that something is put in a certain condition/state and it still is
at the current time. You can see this a little bit like -ている but for inanimate
objects (more precisely, ている indicates the status of a continued
action/movement, while -てある indicates the state of something as a result of
some action/behavior). Also, in this case the state is not being precious but
the being locked/put away.
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しまう has 3 meanings as a verb, 1 usage as a past/complete tense and a usage _do
+ しまう_ unwilling to do.

  1. 物をしまう (formal) treasure / store / put back / 
  2. 店をしまう close / wind up
  3. 仕事をしまう finish /end

    * As a past/complete tense 窓が締まってしまった The window has/had closed (won't be open soon or never.)

    * Unwilling to do 遅れてしまった = 遅れる + する (past=した) + しまう

FYI: 窓が閉まっている the window is closed = 窓が閉めてある One keeps window closed.
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> 母からの手紙は箱の中に大切にしまってあります。

The verb しまう（仕舞う） here is used in the sense of "put away, keep, store
(something somewhere)". It's explained in definition #3 in
[デジタル大辞泉](http://dictionary.goo.ne.jp/jn/100617/meaning/m0u/%E4%BB%95%E8%88%9E%E3%81%86/):

> しまう【仕舞う】  
>  ３ 使用したもの、大切なものなどを元の場所や入れ物などの中に納める。かたづける。「夏物を **しまう** 」「雛人形を **しまう** 」「胸のうちに
> **しまっ** ておく」

～てある indicates a resultant state. しまってある here, which consists of the te-form
of しまう + subsidiary verb ある (or, verb しまう + conjunctive particle て +
subsidiary verb ある), means that 母の手紙 has been put away in 箱 some time ago and
is still in the 箱.

* * *

As to the difference of "intransitive verb + ている" and "transitive verb + てある":

Intransitive + ている -- _current state_ (NOTE)

  * 「窓が閉まっている」 The window is closed. 
  * 「窓が開いている」 The window is open. 

Transitive + てある -- _resultant state_

  * 「窓が/を閉めてある」 The window has been closed some time ago and is still closed. → The window has been / is kept closed. / We keep the window closed. 
  * 「窓が/を開けてある」 The window has been opened some time ago and is still open. → The window has been / is kept open. / We keep the window open. 
  * 「手紙が/をしまってある」 The letter has been put away somewhere some time ago and is still stored there. → The letter is stored / I keep the letter somewhere.

For more, see [this thread](https://japanese.stackexchange.com/q/5505/9831).

NOTE: As you may already know, ～ている can also indicate "continuous/ongoing
action", eg 走っている (is running). And of course we also have "transitive + ている",
eg 本を読んでいる, 店を閉めている.
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