Bear is a formal game term. A character bears the cards played on that character. This usually includes possession and artifact cards representing the items that character is wearing or using, as with Hobbit Sword (1C299) or Mithril-coat (2R105), but it can also include conditions representing circumstances that only apply to that character, like Desperate Defense of the Ring (1R244) or Dwarven Heart (1C10). There are also a handful of conditions, mostly weather conditions, which are played on a site rather than a character.
Except for characters, cards in play are always borne by a character or site, or else they are in the Support Area. A card cannot be both borne by a character and in the Support Area; it is either one or the other, although the bearer may be an ally in the Support Area. If a character is at a particular site, all cards borne by that character are also at that site. Some cards can be transferred from the Support Area to being borne by a character or vice versa, such as Blade Tip (1U209) and all Followers.
Not to be confused with Grimbeorn, Beorning Chieftain (14R6).