When you ascend, you get +1 Hero Soul for each 2000 hero levels you have. If you have beaten Omeet, the boss for level 100, you will get an additional soul. If you beat any Primal bosses, you will receive their bonus Hero Souls as well.
What survives an ascension: Hero Souls, Ancients, and Gildings.
For every hero soul you have, you do +10% more DPS. Also, hero souls are needed to summon ancients and level them up.
Ancients grant you a special power of some kind. To summon an Ancient requires hero souls. To level them up requires more hero souls.
Once you are past level 100, you can earn a gilded hero. You do this by beating a divisible by 10 level for your very first time. That is, beating level 110 for the very first time will earn you a gilded hero. The second time you beat it, nothing. Gilded heroes do +50% more damage than their normal counterparts, and a hero may be gilded multiple times.