There are different grants and thus different deadlines and eligibility criteria, and all grants can only be applied for, when they become publicly announced. Therefore, you can see current grant announcements at the University of Copenhagen homepage:
http://www.ku.dk/sa/su-kontor/legat/aktuelt.asp

Observations: this site is in Danish.

Students: international students in Denmark
Grant-awarding organization: Cirius Danmark, within the Danish government
Study areas: all
Level: all
Grant duration: 2 years
Eligibility: nationals from 27 countries inside and outside the EU
Grant amount: the scholarship covers free tuition plus a monthly allowance of DKK 5,000 (€670).
Application details: you can ask to the national authority (i.e. their Ministry of Education or Ministry of Foreign Affairs)
Application deadline: normally in October or November
Website: http://www.ciriusonline.dk/Default.asp?ID=3771
E-Mail: cirius@ciriusmail.dk 
Observations: Cirius is an agency under the Danish Ministry of Education with the aim to support the internationalisation of education and training in Denmark

Addressees: Students wishing to study in Denmark.
Web page: www.su.dk
Observations: Danish site. You can read this site in English by clicking on the link "In English", on the left side of the page.