Soulbound means that a Familiar or other item is attached permanently to a character. There is no item in the game that removes this bond, and there is no method to remove them in the future.
Soulbound Familiars do not count towards any Familiar tallies. This means that they don't count towards the maximum amount of Familiars, but they also don't count towards Rider prestige requirements. Only certain Familiars can be soulbound. It will say so in the description of that Familiar if it is possible.
Additionally, the Quetzal Familiar can very rarely cause eggs bred by that Rider to hatch with Soulbound Familiars. For simplicity, these Soulbound Familiars are attached to that dragon's Rider (if they have one). If the dragon does not have a Rider--or if it is hibernating--the Familiar will be attached to the dragon instead. When the dragon is transferred, its Familiar will go with it. Familiars of this nature are generated when the dragon is accepted into the group.
Currently, only Familiars can be Soulbound.