Emblems behave similarly to enchantments: They have an ability that, in a general sense, continually affects the game. The primary difference between them is that emblems aren’t permanents and don’t exist on the battlefield. Nothing in the game can remove an emblem, simply because no other spell or ability references them. Once you get an emblem, you keep it for the rest of the game. Emblems have no color, name, card type, or other characteristics beyond the listed ability.

Lethal damage and effects that say “destroy” won’t cause a permanent with indestructible to be put into the graveyard. However, a permanent with indestructible can be put into the graveyard for a number of reasons. The most likely reasons are if it’s sacrificed, if it’s a creature whose toughness is 0 or less, or if it’s an Aura that’s either unattached or attached to something illegal.