I have stayed in Absalon several times, both the Annex and the main. It's old building and hasn't been rennovated since sometimes in the 60's or early 70's one can assume from the furnitures and looks. The rooms are small. In the Annex, the WC and showers are on the hall but a sink in the rooms. WC's are in the rooms in the main hotel. I love the location, few steps from Copenhagen Central station but far enough so you don't hear the trains. The streets around the hotel are a bit shubby, with lots of prostetutes and drug addicts, sex shops etc. The hotel is cheap, so if you want a good location without paying alot it's fine. Staff is friendly and helpful. Couple of times when I booked the Annex I was upgraded to the Main without extra payment, love that.

As other correspondents have mentioned it is a very short walk to the railway station and not much further to Tivoli and the Town Hall Square. It is basic but clean and has everything you need for a short stay. The bedding is light and the pillows tiny but this typical of 3 star accommodation in many European cities, to get "American-style" luxury you have to move up a grade (at considerable expense). The breakfast is also typical European, cold meats, cheese, pastries, cereals, breads and coffee (which was real, but not that great). As a family run hotel it has a nicer "feel" than the big chains and the staff were pleasant and very helpful (and all spoke excellent English).
Parking nearby is a bit of a lottery, and not cheap (even on the street) but at least the ticket machines gave you English instructions (although at a peak charge of $3/hour you should get something for your money!).
As somewhere to stay in Copenhagen as a base for a night or two for one or two people it presents excellent value, but if you have a larger party or intend to spend much time in the room other than for sleeping then it's probably not for you. Especially if you have to look at the decor, in our room everything matched but I remain unconvinced by the bilious shade of green they had selected as the main colour.

Hotel Absalon has a perfect location in the middle of Copenhagen, within walking distance of all major sights and very close to the main station (Hovedbanegaarden).
The staff is friendly and professional, and the rooms are ok.
If you want some extra comfort, you should book a topfloor room; then you actually get the comfort of an extra star, witout paying as much as a four-star hotel in Copenhagen would normally cost.
Please note that Helgolandsgade is a part of Copenhagen's red-light-district, so Absalon is not the perfect place to bring children, if you don't want them to experience that kind of citylife.

The hotel is a 2 - 3 minute walk from the train station.
It is simple and clean enough, though definetly a budget hotel.
Someone on this site had recommended that rather than stay at a hotel near the airport with nothing to do in the evening, stay downtown as the trains run evey 20 minutes and it is a short 15 minute ride. I did that and was very happy that I did.
I don't think I would have stayed at the Annex Absalon for more than one night, but for a cheap (by Copenhagen standard) place and very close to train station - it was perfect.
And the breakfast was excellent!