In general, fruit trees are not grown on their own roots. They are grafted, with the rootstock being chosen for particular traits like size control, disease resistance and soil compatibility, and the scion (which forms the top of the tree) chosen for the fruit it produces.

Apple rootstocks

Standard or full-size trees are grown on seedling rootstock. The rootstocks for these trees are produced from seed, usually from Dolgo, Antonovka, Ranetka, or Baccata fruits. Dwarf trees are grown on clonal rootstocks produced in stoolbeds. In general these produce a less vigorous tree that therefore remains small and fruits sooner.

One of the reasons to avoid purchasing trees in pots in local nurseries is the lack of identification of the rootstock. Trees sold as semi-dwarf with no further identification of the rootstock are usually on M.7, and there are better rootstocks available. (See [[When, Where and How of Fruit Tree Acquisition]])

Fireblight is endemic to Minnesota, and rootstocks should be chosen for fireblight resistance. Rootstocks susceptible to fireblight can be killed by an otherwise minor fireblight infection in the top of the tree.

The ultimate size of an apple tree is determined by a combination of factors:
Cultivar vigor
Rootstock vigor
Climate
Pruning
Soil conditions

My experience has been that most varieties on seedling rootstock won't start fruiting until they've reached a height of 15' or so. With summer pruning, any variety can be limited to 20' in height. Unpruned, the vigorous varieties will reach 35', perhaps more.

Dwarfing rootstocks have been used almost exclusively by the commercial apple industry for decades. Whether they are right for you, the home orchardist, depends on your goals. Their advantages are:
Trees on dwarfing rootstocks bear sooner after planting, generally in 3-4 years.
Average fruit quality is generally higher due to better exposure of the fruit to spray and to sunshine.
Smaller ladders, or none at all, can be used for picking and pruning. This is both a convenience and a safety advantage.
The yield per acre is generally higher.
The smaller size trees are a better fit for smaller urban and suburban yards.

Standard-sized trees have their advantages, too:
The trees are longer lived. Many will bear for 100 years with proper care. In contrast, dwarf plantings generally have 15-20 productive years.
Because the trees are more vigorous, they are better able to recover from mechanical damage (from ice storms, wildlife, contact with lawn mowers or construction equipment, etc) and disease.
Anchorage is better. Dwarf trees are more easily uprooted by strong winds.
Standard-sized trees are generally more tolerant of poor soil conditions – low fertility, wet soil, and sandy soil that doesn't hold water well.
Similarly, standard-sized trees, once established, are more drought tolerant.
Cold hardiness is better, a consideration in Zone 3 and colder areas.
Because of their height, strength, and vigor, they are (once established) less susceptible to damage from deer and other large wildlife.
The up-front effort and expense to establish an orchard is lower due to the smaller number of trees that must be purchased, planted, and established and reduced need for staking, irrigation, and fencing.