When was the Berlin Wall build?

August 13, 1961

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When did the Wall came down?

November 9, 1989

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Is Germany completely unified today?

Yes, but there are several differences between East and West Germany.

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Why was the Berlin Wall built up?

More than 2 .6 million East Germans escaped to West Berlin or West Germany from 1949 to 1961.
Many people had to suffer under repressions of the Communist system or wanted to live a better life in the West.
The East German government saw no other way to prevent from escaping to the West via Berlin than closing the border between East and West Berlin on August 13, 1961.
 
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How long did it actually take to build the Berlin Wall?

The Berlin Wall was build over a very long period and there were different steps of building up the wall.
On the first day (August 13, 1961) the border in Berlin was completly closed and in generally it was not possible for East Germans to set a food to the other part of the city.
East German border guard patrolled on the whole border through Berlin. At first a barbed wire was used but on August 15, 1961 the first Wall was built up.
Until the fall of the Berlin Wall in 1989 the Wall system was extended permanently.

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Why the Berlin Wall was taken down?

At 06.53 pm on November 9, 1989 a member of the new East German government was asked at a press conference when the new East German travel law comes into force.
He answered: "Well, as far as I can see, ... straightaway, immediately."
Thousands of East Berliners  went to the border crossings. At Bornholmer Strasse the people demanded to open the border and at 10.30 pm the border was opened there.
That moment meant the end of the Berlin Wall.
Soon other border crossing points opened the gates to the West
In that night the deadly border was opened by East Germans peacefully.
 
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Have you got any other good links about the Wall?

Sure. Please visit my Link page. 

In some websites, they call East Berlin the GDR and some call it the DDR. Which one is right?

East Berlin was the capital of the DDR.
DDR was the official name of former East Germany and means Deutsche Demokratische Republik (DDR).
The English translation is German Democratic Republic (GDR) and means the same.