A Dynasty tracks your progress throughout the game, both single and multi-player, and allows you to gain some benefits from earlier achievements. It is made up of a bunch of pieces that are all inter-related.
Heroes - Dynasty heroes include your main heroes from the campaign missions, as well heroes you have created for multi-player. Dynasty heroes are able to benefit from other pieces of your dynasty (traits and weapons) and also help build up your dynasty by earning experience and building up your Dynasty weapons.
Weapons - A few items in the game are considered "Dynasty Weapons". These items are available to all your dynasty heroes, and also gain experience whenever the hero wielding them does. They can level-up, giving them additional powers.
Traits - Dynasty traits are some passive bonuses that you can earn. At the beginning of each game, you can select the dynasty trait that you want your dynasty hero to get for that game. You can unlock additional traits by earning achievements, and you can unlock additional slots for traits (allowing you to select multiple ones per game) at the Altar of wishes.
Altar of Wishes - Earning achievements for your dynasty earns you dynasty points. You can spend these at the altar of wishes to unlock new options. Some are cosmetic, like new portraits for your heroes, while others have gameplay impact, like new dynasty weapons or additional dynasty trait slots.
Dynasty Rank - All experience earned by all your dynasty heroes contributes towards your dynasty rank. Increasing your dynasty rank opens up new things to purchase at the Altar of Wishes.