In his "Gettysburg Address," President Lincoln was speaking to commemorate the service and sacrifice of the men who gave their lives during the Battle of Gettysburg, Pennsylvania in the American Civil War. His specific purpose was to dedicate the cemetery where those dead soldiers were buried. Thus, he refers to the battlefield of a contest that had already occurred: the actual battle which was fought at that Pennsylvania town in July 1863, some five months before he gave the speech. But more broadly, he refers to the ongoing conflict that is the Civil War, not confining his message to the violent clashes between the armies but the larger “war” of hearts and minds that, by that point, had been dividing the nation for more than two years.
When Lincoln mentions the “great civil war” in the second paragraph, he refers back to the “new nation” he had mentioned in the first paragraph. This civil war is a test of that nation’s endurance. The fights that are being fought include the underlying concepts of liberty and equality. In the third paragraph, he further emphasizes those concepts and calls the test of them “the great task” that Americans still have ahead of them, the “unfinished work” of reinvigorating the nation. He thus encourages the audience to think of bringing about that rebirth of freedom as their personal battle, not just one that soldiers are carrying out for them.