In the commander tree, if you click upon any of the commanders of an unlocked commander tier, a Reinforce button appears. You can consume artifacts to produce a card of the same faction of the commander. Artifacts required are listed above the Reinforce button. The rarity of the card that can be produced depends on the commander tier.
Rules currently known (or guessed)
With a rare commander, you need 4 artifacts and the card produced is common, with a low probability of being rare. (some say current rate 1 silver out of 25)
The rate of common/rares is 1/10
With a epic commander, you need 6 artifacts and the card produced is rare, with a low probability of being epic.
The rate of rares/epics is 1/20
With a legendary commander, you need 8 artifacts and the card produced is rare with a low probability of being epic or legendary. The artifacts required for legendary reinforce are not currently obtainable.
The rate of rares/epics/legendaries is 1/10/133.3
It used to be possible to get upgraded cards through reinforce at a very low chance.