It has been quite a while since I actively searched the Danish records for my grandfather, his brothers and sisters and his parents, but I was reading the article Back to Denmark by Carol Richey in the July/August 2008 issue of Discovering Family History.
Carol lists the ten most common surnames in Denmark; Jensen, Nielsen, Hansen, Christensen, Larsen, Sorensen, Rasmussen and Jorgensen, and since Denmark used patronymics to about 1850 or later in the farming areas you can have several of those common surnames. I have Hansen, Sorensen and Jorgensen although Jorgensen really should have been Jorgendatter as she was my great grandmother, but Denmark had pretty much gotten rid of the datter part of surnames by the time my great grandmother was born.
Before you can search Danish records you need to find the parish your ancestor lived in. Carol Richey gives several ways, starting with interviewing family, checking local records, obits, emigration and immigration records, and naturalization papers.
I did not use any of those she suggested, I had a notebook my grandfather had used and in the front he had written his address in Denmark, which included the county and the parish. I had also used the Danish IGI at my Family History Center (the IGI is online now) and it listed the parish my grandfather and five of his brothers and sisters were born in.
I will give you a caution about the IGI, one of the brothers of my grandfather was John Hansen and when I typed John Hansen in the IGI I got 14 pages of John Hansen's, and it turned out none was the John Hansen I was looking for. I did find him later on in the IGI, he was listed a Johannes Hansen.
Carol Richey lists using the Danish Census and the census is a good resource and even if you do not read Danish very well it is easy to understand. Another resource for males is the military registration lists, Danes registered the males at birth or soon after birth. They had a new registration about every second or third year and the father was required to register all his male children. If the child was born the year of the military registration it listed the birth date, and his fathers name was listed on all the military registrations. If the child was born in an off year for the military registrations it just listed the age of the child.
The birth records in Denmark listed the child's birth date, his parents name, parents occupation, parents place of birth (the parish name) and the age of the parent. The wife kept her maiden name in all those records, so it was easy to find her parents, so then you can go to that parish and check out the records for the parents. Those records listed the grandparents and so back another generation. I found all these records in the Family History Library at Salt Lake, but today many of them are turning up on the internet.