Denmark is part of the European Union's Schengen agreement, which means that you do not need to show your national ID card or passport when you are travelling to or from Denmark from another Schengen EU country. You are still recommended to bring your passport or ID card with you in case you need to prove your identity. Some airline operators still require you to present a passport even for travels within the Schengen area.
The following countries are part of the Schengen Agreement: Austria, Belgium, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden and Switzerland.
EU citizens can travel freely to Denmark; citizens of other countries may require a visa. If you travel to Denmark from outside the Schengen agreement area, you may also require a visa. See here if you are from a country where you will need a visa to enter Denmark.
There are certain restrictions to bringing pets and other animals in to Denmark. Consult the Danish Veterinary and Food Administration website to see what rules exist regarding your pet.