In the Gettysburg Address, Lincoln says that "it is for the living rather to be dedicated here to the unfinshed work." What is this unfinished work?

In order to understand what Lincoln is saying in this
line, you need to look further down in the speech.  Lincoln goes on to tell us what the
"unfinished work" is.  Basically, the work that the living must continue to do is the
work of making sure that democracy can exist.

In the
speech, Lincoln says that the Civil War is a war for democracy.  He says that it is a
test of whether a government that is

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... conceived in liberty, and dedicated to the
proposition that all men are created
equal

can last for any long
period of time.  The men who have died in the battle, then, have given their lives to
try to ensure that democracy can last.

According to
Lincoln, the living need to work to preserve democracy.  They need to continue where the
dead left off.  If they do they will be able to
ensure

that
government of the people, by the people, for the people, shall not perish from the
earth.

So, the "unfinished
work" is the work of keeping democracy alive and well.