You can set up as a sole trader, a partnership, a private limited company or a public limited company.
A sole trader (enkeltmandsvirksomhed) means that the finances of the business are also your private finances (though you should keep the business accounts separate). If the business makes a loss, then you as the owner and a private person are not protected. This means that everything that you own as a private person, can be used to cover losses in the business and that includes your house if you own one.
The conditions are the same in a partnership (interessantskab) except that in this case you are working with one or more partners and have signed a partnership agreement that covers for example how much each partner own of the business. Your company name will end with the letters I/S.
In a private limited company, this means that you as a private person and the company are seen as completely separate entities according to the law. So, if your company makes a loss, then people and organisations which are owed money by your company cannot make a claim on your private assets to recover that money. Your company name will end with the letters ApS to warn anyone doing business with you that they can only recover assets from the company. You are required to put in at least 50,000 Danish crowns to start this type of business.
Note that a special type of start-up company format called iværksætterselskab, IVS, has been abolished and that it is no longer possible to set up a limited company in this way. IVS was a cheap way of setting up a limited company requiring only one Danish crown to set up.
You can find more information about the different business types in English here: https://www.startupsvar.dk/registration-of-business