While the war seems to have started largely as a reaction to the new Danish parliamentary action bringing about the end of Holstein's regional autonomy as part of a general German nation, actual military campaigning did not lead to much until the middle of January of 1864. Originally the Danish forces and the assorted German forces were not involved in any decisive movement, but positioned in the different banks of the Eider river. This changed through Bismark's first plotting to secure a part in the war for Prussia and Austria which would pretty soon exceed the wishes of the German polities west of Prussia. War was accordingly declared by those two powers.
The Danish command was at first set (it seems also for historical reasons) to defend along the Dannevirke, an old Viking creation of a rising barrier, reaching up to 6 metres, and not falling considerably below 4 in any parts of it. The Dannevirke was expanding for around 30KM, and consisted of a small segment in the southern entrance to the land just below the river Shiel, and then a larger one reaching towards the Wadden Sea.
The Danish army command and the new King (who signed the parliamentary action), decided to leave Holstein and focus on defending that line, mostly because the front would therefore be considerably smaller, and also counting on the various water and river patches to the east of the front, which would on the one hand make it harded for the Prussian and Austrian armies to cross, on the other be further secured by the relatively better armed Danish navy.
But in the end the Dannevirke was also abandoned, since the Prussians led their army in the midst of a horrible winter (temperatures were routinely lower than -10 Celcius), which turned those bodies of water into an icy bridge for the critical first part of the war.
It is argued that the retreat and abandonment of Dannevirke (which never again returned to Denmark) without a battle, played a very crucial demoralising role to the Danish army.
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While the army of Denmark numbered around 40K soldiers, and 100 guns, along with a decent navy including the fully armored ironclad Rolf Krake, the Prussian army was in the onset of a war a little smaller, and along with the Austrian army numbered around 60K soldiers, with nearly 160 guns. During the war a number of heavy artillery pieces were captured (particularly due to the abandonment of Dannevirke where many were installed and could not be moved to the next line), and 20K more soldiers reached the ranks of the combined Prussian-Austrian army.
The Danish army was ordered to retreat to Dybbøl, and along with Als those were the final areas where it was sieged. The actual campaign after the voluntary retreat from Dannevirke seems to have been one decided as of its outcome in the moment that Bismark took advantage of this event so as to enable an incursion into actual Schleswig. It appears that the Danish side did not expect this development, and instead was hoping for either a negotiation about Holstein's status, or a prolonged stalemate, which could lead the main European powers to intervene (as they had done in the first Schleswig-Holstein war). However none of those happened, and ultimately the sieges were practically impossible to break by the Danish side. The peace signed in the end of october, almost 10 months after the war started, saw Denmark losing 40% of its territory, along with 35% of its population. Also it ceded most of its railway network, and important towns such as Kiel and Schleswig. The war in the long run secured Prussia's position, and 18 months later Prussia won against Austria, with the help of French and Italian threats against Tyrol and Dalmatia, paving the way ultimately for 1870...
It begins in 5 days 
