From the normal meaning of the words I would have assumed that items that become bound to you on equip are unremovable, but that doesn't seem to be the case. I can take off bound armor, and I can sell it to the in-game vendors. So I have two questions:
Bound items are a feature implemented in MMORPGs to avoid abuse of twinking.
Bound items are designed to prevent them from being shared with other characters, to avoid easily making overpowered lower level characters. If you have a bound item, it can (generally) be sold to in-game vendors, but it cannot be traded to another player.
Unbound items, however, can be freely traded from player to player, and can be used by players who aren't the ones who initially obtained the item (whether from quest, random drop, or by purchasing it from a vendor), assuming that the player meets all of the other requirements for the item.
Bound and unbound is not random. Quest items are typically bound, as they are the reward for that character completing the quest, and therefore it is expected that that character will be the one to use it. The same applies for specific reward drops from difficult encounters, such as raid or group instances.
A bound item is something that cannot be traded to another person.
To answer the second point, it is based on the item. Many boss drops will be "Bind on Pickup" That means whoever wins the roll will have that item bound to them, they cannot trade it, they can only sell it.
Other items, will have a "Bind on Equip" tag. This means that they can be freely traded until someone equips that item on their character page. Afterwhich the item is then bound to the player and follows the can only be sold/reverse engineered.
Bound items cannot be traded with other players, but can be removed, stored, or upgraded for your character like any non-bound item.
They are a staple of most modern MMOs (and some that aren't, like Diablo II), and they prevent players from easily transferring power from one character to another. Level or stat requirements are in a similar vein.
This is answering one facet of the question as the other facet seems to be answered sufficiently.
Going on the system used on other MMOs:
"What determines whether an item will bind? Is it random, or arbitrarily chosen by BioWare per item?"
This can be broken down into three questions:
The first question would cause any gear not bound on pickup to be farmed for credits. If particular NPCs were found to drop the item they could be camped by same said farmers.
The second is pretty intricate to show real numbers on, but, linked to the above, those farmers would gain impressive enough wealth to influence the market above and beyond what is safe for one entity to have.
Thirdly, if everyone has the same gear we would all walk around looking alike, "Attack of the Clones"!!!! Making something BOP stops this from happening to an extent.
