Bind on Account, or BoA designates an item that "binds" to the account of the character who first picks it up. These types of items could be passed between different characters on the same account (presumably by using the shared stash), but could not be traded to another other characters.

BoP, Bind on Pick-up is essentially identical to BoA, in function.

In Diablo 3?

It is not confirmed that there will be BoA items in Diablo 3, but the D3 Team has talked about using BoA as a way to keep the end game economy functional. The theory is that when top level items can be traded, eventually every character has the best of everything, since characters who indulge in huge amounts of item finding (or illegal duping, in the case of Diablo 2) will score enough of that item to distribute them widely.

Compare BoA to BoE -- Bind on Equip which is also known as "soul binding." These types of items bind to the character who first equips them; they can be traded, perhaps only within the same account, but once any character actually uses the item, it is bound to that character permanently. (Characters can dispose of such items for space considerations, but they can not give or trade them to other characters.)

Blizzard Comments

Jay Wilson addressed this issue in an interview from BlizzCon 2009. [1]

Jay Wilson: We have no “Soulbound” or bind-on-pickup, except for quest items. We do have bind-on-equip for the highest end items in the game. We targeted, roughly, any item above level 85. These we will do as bind-on-equip. The reason for this is that we want people to be able to trade them, but we also want to remove the high-end items from the economy. One of the greatest ways that you can do that is with bind-on-equip. What we don't want is to have a situation where you find something on the ground like, “Oh, man. This would be a perfect weapon for my Monk. Oh, but I just picked it up and now it's on the wrong character.” We don't want that at all.

Most of our focus on Diablo 3 is as a trading game. So, if you take trading out of the item space, you ruin the core of the game. Finding a really great item that is not for you is still a great event because it means you have a bartering tool to get the item that you do want. We definitely want to make sure that that still exists.

Bashiok addressed this issue during a mini-interview posted as part of Blizzcast #13, in February 2010. [2]

Zarhym: Will all Bind on Equip or Bind on Pickup items be bound to account? So you can hand them down to your other characters as you find better gear.

Bashiok: Maybe. It sounds kind of cool. So I asked Jay and some of the other designers about this actually because I wasn’t sure, but Jay’s response was “That sounds awesome!” And he likes things that are awesome. Which is a very Jay Wilson quote. But there could be economic reasons that we don’t do that, there could be gameplay reasons we don’t do that. It’s probably too early to say, but that sounds cool so we’ll have to see.

Bashiok elaborated on this later that same day, when a fan pointed out that BoA could lead to stagnation in the economy, if every account eventually had several of the best items that players could pass around between their different characters. [3]

Totally, which is why I said “But there could be economic reasons that we don’t do that.”

It would probably stagnate the economy… UNLESS, there are other systems in place that could make it attractive for players to essentially destroy their BoAs. So I don’t think it’s without solution at least, but it could get sticky if not balanced or pulled off just right.

Anyway, we’re not really seriously considering BoA items at this point in time.

References

August 2009: Game Director Interview by G4TV.com.
February 2010: Bashiok on Blizzcast #13.
February 2010: Bashiok forum post.