I would like to root some apple branch cuttings. When should a cutting be taken and is there an optimal part of the tree from which to harvest the cutting? What is the best medium in which to attempt rooting, is rooting hormone helpful, what other conditions are required and how long should it take to successfully root? 
Carl
I've never seen it done. It isn't widely done. The only fruit trees where I have heard of own-roots propagation being done on any sort of scale are Antonovka apples (which are prized for their taproots) and Evans cherries (which do not graft well onto any of the customary rootstocks due to compatibility problems). Both of these are propagated by digging up a piece of root and dividing it.
I would try rooting hormone and warm temperatures if you want to give it a try. I've seen some reports stating that some cultivars are nearly impossible to root in this fashion while others are no more difficult than most other trees (which is still fairly difficult).
Grafting is generally the preferred method of propagation.
…I’ll agree with FrozenNorth… Company arrived just as I was completing the following, and no one had answered you yet… here’s what I had:
Apple trees are rarely started from ‘cuttings’ and I have little knowledge as to how it might be done. What comes to mind as a ‘down-side’ is that you’d end up with a tree(s) anchored by its own root system; some vigorous, if in most cases ‘too vigorous;’ most susceptible to a multitude of diseases; and likely lacking the ability to withstand varying soil conditions, such as ‘wet feet.’
I suspect varietal vigor would determine the ultimate height; with a Gravenstein or Transparent heading toward 40 feet - whereas, from my experience, a Braeburn or Wolf River ending up at a lesser height.
Also keep in mind, like most highbred cultivars, the resulting fruit bearing ‘wood’ is not nearly as resilient as ‘wild stock.’ Much in-breading has occurred, varieties crossed with relatives; thus their weaknesses are magnified. Rootstock is much closer to ‘wild,’ thus better able to withstand the pathogens and stress conditions underground.
Though potted rootstocks are likely difficult to find this time of year, summer budding or dormant grafting to a known rootstock would give a more predictable result and a far longer lived tree than a self-rooted cutting. When I graft a tree I expect I’m handing over something with the potential to last a couple hundred years. If I were to start a tray or bed of rooted cuttings… I would not expect the same …and be very reluctant to pass them around.
…now if you’re still set on starting apple trees from cuttings, I suspect we’ve got guys that can likely walk you through the process. I know it ‘can’ be done, but as mentioned, rarely is.
If you are set on rooting the apple you might look into layering or air layering. With air layering you wound a branch, apply rooting hormone, and put a container of moist peat moss around the wounded area. The container needs to maintain moisture. Roots will form after a while. I've done layering with filberts and other trees: bend the wounded branch down to the ground and cover with soil, you may have to stake it down or put rocks on top of the branch to hold it down. After one year you can cut off the branch and transplant.
FrozenNorth, Viron and Dubyadee, thank you all for your suggestions. I thought of the root-harvesting method too because I harvested a volunteer root shoot once and then grafted onto it. That tree is bearing apples this year. I'm not sure how to expose a root and cause it to give off a shoot though. I like the idea too of bringing a branch down into the soil while still attached to the parent tree and waiting until it grows roots. I will try both methods. 
Carl
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