Short answers, pulled from the story.
The Second Schleswig War began on the 1st of February 1864, when Prussian and Austrian forces crossed into Schleswig. It ended on the 30th of October 1864 with the Treaty of Vienna, in which Denmark ceded Schleswig, Holstein and Saxe-Lauenburg to Prussia and Austria.
Denmark fought against the combined forces of the Kingdom of Prussia and the Austrian Empire, both acting on behalf of the German Confederation. The Prussian-Austrian army was commanded overall by Field Marshal Friedrich Graf von Wrangel, with Austrian troops led by General Ludwig von Gablenz.
The war started after Denmark's November Constitution of 1863 tied the Duchy of Schleswig more closely to the Danish kingdom, a move the German Confederation viewed as violating the London Protocol. Bismarck issued an ultimatum on the 16th of January 1864 demanding the constitution's repeal within 48 hours, and when Denmark refused, Prussian and Austrian troops invaded Schleswig.
Denmark ceded the Duchies of Schleswig, Holstein and Saxe-Lauenburg to Prussia and Austria under the Treaty of Vienna, keeping only the island of Ærø, the town of Ribe and eight parishes south of Kolding. The kingdom's land area fell by 40 percent and its population dropped from 2.6 million to about 1.6 million.
The Danish army abandoned the Danevirke on the night of the 5th of February 1864 after a hard winter froze the water and marshes protecting its flanks, letting German troops bypass the line. Commander-in-chief Lieutenant General Christian Julius De Meza ordered the withdrawal to Flensburg, and he was relieved of command soon afterward.
In Denmark, the defeat was considered a national trauma that pushed the country away from using war as a political tool until the 1999 NATO bombing of Yugoslavia. In Germany, the victory helped Otto von Bismarck win over Prussian liberals and contributed to the later rise of a personality cult around him after his death in 1898.