Note; this is not about whether a game is "always online" or requires an Internet connection, nor whether it requires the dev's servers to be online. It's only about whether real-time interaction with other players is available/required.
E.g. Sea of Thieves is an online-only multiplayer game. It has single-player as well as multi-player.
Whether any achievements are dependent on a difficulty level - e.g. "Complete the game on Veteran difficulty"
Note that if a game has cheat-like difficulty options in the settings (e.g. disabling deaths), and doing so invalidates all achievements that does not count as a Yes here. Still a No.
Achievements are first categorised into different kinds, based on what player action is required to unlock them. Note the distinction between Objective, Grind and All-the-things achievements.
Story: Achieved through progressing the main campaign/store (see above). Many Story achievements are unmissable, but some may be optional, missable or mutually-exclusive based on story choices.
Grind: Requires the completion of multiple identical tasks (e.g. "kill 500 enemies"). Progress typically cannot be reset/rewound, so achieving the goal is a case of grinding out the activities throughout play. Usually there are more opportunities to do the thing than are required for the achievement (i.e. there would be more than 500 enemies total).
Objective: Requires the completion of a singular objective; e.g. completing a quest, reaching a location, etc. Note that some grind-like achievements belong as objectives if progress toward them can be reset through gameplay; e.g. Forza Motorsport's "Safety Superstar" achievement which requires the completion of 10 consecutive multiplayer races with an S-class safety rating. The 'consecutive' obligation, and the reset of progress if your safety rating drops below 'S' make this essentially an Objective rather than a Grind.
All the things: Collectibles, and other kinds of "do every kind of X" type achievements such as "craft every type of weapon" or "discover every fast travel point". This also includes "kill all" achievements where each of the things to be killed is a unique enemy. Achievements of this type should normally have a checklist included in the guide so you can easily tell which of the things you have done already and which ones remain.
Missable: Can only be achieved in a certain part of the game, and not fulfilling the requirements will mean you can no longer unlock it without replaying the game or loading old saves.
Soft-missable: Can only be achieved in a certain part of the game, and is missed then some replay is needed to unlock (e.g. replaying a mission), but without needing to save-scum, chapter-select or restart the game.
Unmissable: Cannot be missed as part of natural toward completing the game.
Optional: Can be achieved at any point through the game, up until game completion. Note that whether an achievement can still be unlocked after completing the game is separate; the games's overall/roadmap section should describe any post-completion limitations.
Mutually exclusive: Cannot be achieved at the same time as one or other achievements. I.e. you'll have to make a choice, and whichever outcome you don't choose, those achievement(s) become missable.
Whether an achievement needs other players to be able to unlock. This can be an important flag as multiplayer achievements may become hard (or impossible!) to unlock for older games with few remaining players - or where the developers have turned off online services.
Games that are purely single-player-only or multi-player-only will not display this flag; as all achievements would have the same value.
Anywhere: Can be unlocked in single and/or multi-player modes.
Single-player: In offline/single-player mode, no interaction with other players required.
Multi-player: Requires an online multiplayer mode, and/or the online synchronous interaction with other players (e.g. an Invasion mode).
No known glitches: No problems are known. Achieve away!
Glitched: Known to be problematic and not unlocking - either for some people or for everybody. This is due to a game bug, and so there should be hope that a future patch will resolve the glitchiness.
Previously glitched: Was glitched in the past, but has been fixed by a subsequent game patch.
Unobtainable: The game has changed, via a patch, such that the achievement is deliberately no longer achievable - e.g. the removal of Arena mode in Sea of Thieves resulted in lots of Arena-related achievements that are no longer obtainable. Unobtainable achievements typically will never be obtainable in the future.