thank you for you question.
the 2 main rules/possibility shortly explained:
The best for you would have been if you applied for membership in a "A-kasse" more than 1 year before your end study.
Membership is free of charge for students, and if you have been a member for at least one year before ending study you can have money from day 1 after ending study. You will get "Dagpenge" from the Danish State in case you did not get a job.
There is a second rule/possibility:
If you have not been member of a "A-kasse" during your study or leass than 1 year, you can get "dagpenge" from 1 month after you ending study.
Remember to apply for membership at latest 14 days after ending study.
You can find relevant A-kasser here: http://a-kasser.dk/studerende/
From here choose your education in the drop-down. Then you will be directed to the relevant A-kasse for you.
As long as you are student all A-kasser is free of charge.
If you later have job you have to pay for membership.
You can then find the cheepest one at our site www.a-kasser.dk.
You have the same rights no matter what A-kasse you choose. Because it is the Danish State who will pay you in case of unemployment. BUT is is important to be a member of a A-kasse. Otherwise you will only have the right to receive a very little amount ("kontanthjælp").
Hope it helps..