There's a disputed succession in the Duchies of Schleswig and Holstein, which he ruled, between his Danish successor, Christian of Glücksburg, whose wife is the heir-general to the old King, and Frederick of Augustenburg, the heir-male. Augustenburg's claim is that while Denmark itself can pass through a female line, Schleswig and Holstein cannot, and ought to pass instead to him.
Matters are complicated three days after Frederick VII's death, when the Danish Parliament approves a constitution which unites Schleswig to Denmark. Although the great powers had all recognized Christian's right to succeed to the duchies as well as to Denmark itself, this was clearly in violation of international law, because the First Schleswig Crisis had been resolved by supposedly guaranteeing the special rights of Schleswig.
Augustenburg, meanwhile, garners a lot of sympathy and support from the German states, who want to give the duchies over to him to create a new medium-sized German state in Schleswig-Holstein.
The Austrians aren't at all sympathetic to this, but, as leaders of the German Confederation, can't allow Denmark to simply annex Schleswig without protest. In OTL, Bismarck pretends to agree with them and guides both countries into a war with Denmark over the Duchies. By allying with Austria, he is able to ignore Augustenburg's claims and finally break with Austria, defeat them, and annex the duchies himself (along with much of the rest of northern Germany).
I can't remember who it was, but as to the Schleswig War, some British? minister said: "Only three people understood the Schleswig problem: a German professor - but he went mad. The Prince-Consort [Albert] - but he's dead. And myself - but I've forgotten all about it."