Sony has detailed how Bloodborne's online multiplayer will work. In many ways it's similar to that of From Software's Demon's and Dark Souls games, but there are some subtle differences.
The most striking of which is the players can set a password for their game session, then only players who enter the corresponding password can enter their game. As such, you can create a private match just for you and your friends.
This option is only valid in co-op, not versus modes.
Other than that it's pretty similar to the Souls games. The host summons allies with a Beckoning Bell item, up to two other players can respond to their call with a Small Resonant Bell, then the party tackles a boss in the host's game until the host either defeats them or falls in battle.
For Versus games, the host player must be in an area with a "bell-ringer woman." They must then either initiate a co-op game or use the Sinister Resonant Bell to summon the aforementioned bell-ringer woman. The "guest" (invader) uses the Sinister Resonant Bell as a way of invading another player's game if there are any opportunities to do so given their current location.
Versus play continues until either the host or invader kill each other, or the host makes their way to a boss, which kicks the invader out of their game.
Messaging has been changed a little since the Souls games as players can now vote on whether they find certain messages "fine" or "foul."
Replays of other players' deaths is a returning feature. Fallen players will leave tombstones rather than bloodstains, but it's only an aesthetic difference. You'll still be able to watch specters of other players' final moments. You'll also occasionally encounter ghosts of other players exploring the same area as you. You can't actually interact with the ghosts, but watching them may prove educational.
You will need PlayStation Plus to take advantage of the game's various multiplayer features as well as upload the procedurally-generated Chalice Dungeons you may come across.