Exclusive items are items that cannot be transferred to another character by any means (most Exclusive items can be sent to a character on the same account, so they are "exclusive" to your character's account) and can be identified by a small green dot in the top right corner of the item information screen, with the letters "Ex" in it. They cannot be traded, bazaared, or put up for sale on the Auction House; however, some Exclusive items may be sold to NPCs, though it varies from item-to-item whether or not this is possible.
Exclusive items are usually obtained by either completing a quest, defeating a monster, or from examining a checkable point (such as a ???), though a small portion of them can be bought from NPCs. Exclusive items are often used in quests to ensure that the character actually goes through the trouble of obtaining the item themselves, rather than buying it off of someone else.
If an item is Exclusive, but not Rare, a character can possess more than one of the item at a time. Some Exclusive items may even stack, such as Beastmen's Seals, though it depends on the specific item.