What did the presidential election of 1860 show?

The main thing that the presidential election of 1860 showed is that the US had become badly split along North-South lines -- enough so that it would be practically impossible for the two regions to be reconciled with one another.

Abraham Lincoln won the election, but he did so without the benefit of any Southern votes.  When Lincoln was elected in this way, the South felt that their wishes would be completely ignored by Lincoln.  They also felt that this showed that they were essentially powerless when it came to national politics.  Once they came to feel this way, they believed it was time to secede from the Union.

So, the election of 1860 showed that the Union was likely to split and that the Civil War was more or less inevitable.