Added note: all stat increases are gathered from personal comparisons from 2 different saves between the stock at fully reforged equipment. As with all opinions, they differ between person to person, so the best choice is to make your own observations ;3

While not a secret, the alternative skins and improved stats of urfael, Azkar and Acharn arent explained much within the game. To upgrade, or reforge, these weapons their legends must be completed. For every 5 legends of that particular weapon is completed, it obtains a new appearance and slightly improves in speed and damage.
Urfael will swing faster and harder, Azkar will draw more quickly while dealing more damage and Acharn will perform its stealth assassinations, brutalisations and brandings more swiftly.

Urfael's first upgrade, as well as slight increase to stats, changes to:
And eventually upgrades into:

Unlike its sword and dagger brethren, Azkar can only be seen in the wraith world and does not alter much in design, even when fully upgraded. however, it still improves statistics as well as adding inscriptions when viewed on the rune tab.

The first reforge of Acharn is as follows (thanks suPirman :3):


The final reforge of Acharn represents less of the broken blade of Talions son, and turns into:

While the finished weapons provide the best stats, due to the varying designs players may wish to complete only complete some or non of the weapon legends.