As already was mentioned, the game will support SP, Co-op and MP modes (they're mentioned in their priority here). The singleplayer will behave the same way it does in the original game, so there's almost nothing to talk about, but the other two definitely requires some clarifications.
From a technical viewpoint, there are no issues with supporting multiplayer in a "game designed singleplayer-only", it might be just a bit immersion-breaking in some aspects (like players suddenly appearing in front of you and such), but the game will remain playable, despite that
There will be support for up to 4 players to play the original (singleplayer) campaign. The game rules will be tweaked to scale to players count (by multipliers) so monsters HP and loot amount will depend on how much players are playing (1 player = original difficulty settings, each additional player will add 50% to monsters HP and (25% to) loot amount (at least for now) so for 4 players each monster will have (original game settings for the selected difficulty level) + 150% = singleplayer HP * 2,5 which should be enough to deal with)
All players will be in one "party" and will be able to freely travel across the whole map, talking and trading with NPCs and kill creatures, but only host will be able to start quests (at least for now). If you kill an NPC related to some quest, the quest will be failed. You need to stay close to host player when he's about to start a dialog with a quest giver NPC in order to accept the quest too, otherwise, the quest won't be started for you and, upon completion, you won't receive a reward
All players info (player stats/info about completed quests/etc) will be stored on a server, so players won't be able to hack their stats locally and connect to someone's game with 1000K of stimpaks or something like that. You start an SP game - all stats will be saved to your game's save file, you connect to someone's game - your stats will be saved to the server's save file. This way it will always be fair because players won't be able to modify their stats/inventory locally or "server-hop" in attempts to find good loot
You're not forced to play in co-op/multiplayer, you will always be able to start a 1 player local server to play singleplayer, and, if you're playing in co-op with someone, that ruins the gameplay experience or annoys you, it will be possible to kick/ban him from the game
That's only a draft version for now, but this is how it's planned to support co-op mode in the game at this stage. If you have some concerns/suggestions about the mentioned rules, leave a comment
Multiplayer support will be mostly oriented on mini-games rather than a freeroam with no NPC on a map. There will most likely be Lua support for server gamemodes (VaultMP returns) so other people will be able to write gamemodes which modifies the game rules and provide some interesting ways to play the game with other players (examples: Capture the Flag, Reach The Point in Time, Protect the Area from Ghouls/Deathclaws, Survive for X Minutes Fighting Against Ghouls/Deathclaws, etc). For now, it's planned to support only 8 players (hard-coded at the engine level), but that might potentially be changed to be controlled by the server. Once again, it is planned to support gamemode scripts written in Lua, so some of the game rules will be controlled by the gamemode scripts (even co-op support most probably will be implemented this way), so it will depend on the server how the game will behave. Lua scripts will allow a lot of customization options.