How can we trust the Bible after it has been translated to another language? Islam considers the Qur'an to be authoritative only in its original language. Many teachings even claim it is a sin to read in any language but it's original, and while not all adhere to this belief, learning Arabic specifically to read or recite the Qur'an is common even in countries where it is of no other use.

On the other hand Christians openly encourage translations into other languages.

What does Christianity believe about the Bible that makes this possible? Are translations generally considered authoritative in their target languages in the same way that the original text was?
The different between the Quran and the Christian Bible is that the Islam teaches that the Quran was dictated in Arabic and written down - in only one language and thus saying and reading it in this language is deemed more valuable.

The Christian bible was written by men, inspired by God, having their own style, grammar and vocabularly. i.e. the greek of Luke (who was an educated man) is quite different to the greek of Mark (who wasn't educated).

Furthermore, the bible was written in 3 languages, Hebrew, Aramaic and Greek.

The new Testament writers quoted from a translation of the bible. The Septuagint is a greek translation of the old testament which the N.T writers used as their old testament scriptures, so they were already using a translation.

So it is more the message of what is being said rather than the words themselves that are important. 

Yes, some details and nuances of the original are lost in translation, the general gist and message can still be conveyed accurately and authoritatively.