I haven't discovered every last secret in the game, so I can't claim to be authoritative, but based on what I've experienced in this game so far and from my experience with prior games in the series:
- You can't participate in any kind of co-op or PvP, obviously.
- You can't read or write messages, except for messages left by the developers.
- You can't meaningfully participate in (or earn rewards from) any convenants that require PvP or co-op interaction to do so.
- Certain boss fights or protected areas that are designed to bring in other human players as opponents will probably use AI-controlled NPCs in those players' place.
- All of the story/NPC elements are almost certainly still accessible, as NPC progress is solely based on your solo exploration and interactions, which are mechanically unchanged in offline mode. This is also how it worked in both previous Souls games.
- The sole exception to this may be any covenant NPC dialog or events that are only accessible after you've ranked up in one of the aforementioned co-op or PvP-oriented covenants.