http://en.wikipedia.org/wiki/Close-mid_front_rounded_vowel This is the vowel that ø represents. It doesn't appear in standard English. I would say the closest approximant is the "oo" sound in "book".
I find it interesting that (in the android app at least) they never clarified that the d in words like brød and vanden is actually pronounced more like a soft gutteral L than anything resembling a d :o I was mystified so researched further, and apparently it's up there with the most difficult letters to pronounce in Danish!! Yet it's introduced (with no explanation at all) from the very first lesson!!! I'm loving the course so far, but really wanted to point that out. I wonder if I'm the only person who struggled with that for a while! :)
Hi there. I don't know when you wrote your comment, but I got caught on that too. Look on YT for videos about 'danish soft D' it's not a sound that exists in English so you have to learn to make it!! In my head, brød comes out like 'brolt' and brødet 'bro-ld' that helps me remember the correct pronunciation. But yeah it's not a sound in English so go on YT and watch Danish people explain the sound for you, it really helps!! :)