Heirloom gear is collected from chests after defeating some bosses. Your character will inherit any heirloom gear collected when they start a new game. This gear has no enchantment such as elemental bonuses, but it will increase in level along with the character up to its maximum level. For instance, if the character inherits a level 10 helm and they start at level 1 the helm will be level 1. When they advance to level 2 the helm will increase in level as well. This will happen up to level 10.
Heirloom gear is meant to be sold off when you find something better and is just there to make starting at an advanced scene possible. There are heirloom versions of each gear slot and the max level that a character has in each slot is recorded on the Heirloom tab.
Heirloom gear is saved as soon as it is found, whether the character uses it or not, and will persist even without prestiging.