The Silence are a religious order that tried to kill The Doctor because of a specific prophecy which stated,
On the fields of Trenzalore, at the fall of the eleventh, when no living creature can speak falsely or fail to answer, Silence will fall when a Question will be asked, a question that must never, ever be answered.
Dorium Maldovar told the Eleventh Doctor that the question ws 'Doctor Who?' which he had been running from his entire life. He further went on to say,
The Silence are determined the question will never be answered. That The Doctor will NEVER reach Trenzalore.
The Doctor is surprised to discover this
And silence would fall. All the times I've heard those words, I never realized… it was my silence. My death. The Doctor will fall.
That being said, with Doctor Who, we're never sure how to interpret any of this. We know he will eventually be on the Fields of Trenzalore. We know that it is very very possible that Clara will find his real name in the library in the Tardis at some point. We also know that NONE OF THE DW FANS WORLDWIDE want to KNOW the "name of The Doctor" and we would REALLY REALLY love it if silence really would fall when the question is asked. But do we want this enough for him to die ? Of course not !!
Then again, the fall of the eleventh, could mean anything. It could mean morally, or ethically or in so many other senses of the word "Fall". Trust Moffat to use a word like "fall" with so many meanings.