What variety of apple tree is this
From Dr. Jim Luby, U of MN faculty:
We can not positively identify a cultivar visually though we can sometimes make a guess. The history of the plant is very important and unfortunately, people often don't know it. However, you can read about the plant especially if they know it's a University release and narrow down your fruit ID to 2-3 cultivars based on size, color, fruiting time. etc. Here are some resources to help:
U of M varietal releases: http://mnhardy.umn.edu/varieties/fruit/apples/all-apple-varieties
Extension Yard & Garden Fruits: http://www.extension.umn.edu/garden/yardgarden/fruit/
Also, here is a nice discussion about identifying apple varieties. Though it is written about English varieties, all the principles are universal.
If you want to pursue more conclusive testing, UC-Davis will accept samples for DNA fingerprinting. Cost is $345 per sample.