It's fairly simple, actually - the original Dakota territory's capital was moved from Yankton to Bismark, ie South to North, in 1883. This lead to some disagreement among Southern Dakotans, who applied for statehood - only to be turned down.

Soon after a more sophisticated attempt to make a South Dakota was launched, in which a state constitution was drafted, a slate of government officers were approved and senators were chosen. However, once again, Washington refused statehood.

Finally, in 1889 enough pressure was placed on the federal government that the North-South split was finalized.

However, in an interesting little story, another squabble arose - which state would be admitted first? Which would be the 39th and which would be the 40th? This lead to political bickering until President Benjamin Harrison came up with a clever solution:

At the signing of the proclamation admitting the two states, the President put each document inside identical newspapers, with just enough of the statehood documents exposed for the president to sign. He then shuffled them back and forward until no one could tell which paper belonged to which state. He then signed each and removed them from the paper. Due to this it's still unknown which state is 39th and which is 40th (however it is worth nothing that North Dakota is usually listed as the 39th as it comes before South Dakota when listed alphabetically).

Source: Shocking Secrets of American History by Bill Coate