A Hidden role or ability is a part of the player's role that the player is not aware of, sometimes called a "mod note". It is the opposite of a player being self-aware.
A Hidden role is a role that exists as part of a player's role, but is not shown in their role PM. The role functions entirely normally except that the player is not told about it. For example, a Hidden Bulletproof Townie would be immune to dying at night, but would not be aware that this was the case; their role PM would look the same as that for a Vanilla Townie.
Hidden roles are always passive, because a player who is unaware that they have a role is unlikely to try to activate it.
Hidden is the opposite of False: a False role is shown in a player's role PM but actually doesn't exist, whereas a Hidden role is not shown in a player's role PM but actually does exist.
A hidden aspect of a role does not need to be an actual role; for example, it could be an unknown "side effects" of active abilities (or on occasion what an active ability does at all!), or an unknown or unusual action resolution. Hidden Win Conditions are considered bastard modding.
Players may or may not receive hints that their role has hidden aspects. If a player is not aware of the possibility that their role may have a hidden aspect, or if their stated role is not their real role, it is generally considered bastard modding. There are some cases which are more widely tolerated than others (e.g. Cop sanities in the past – although nowadays, a non-Sane Cop would come as a large surprise to most players – or cases in Theme games where the possibility of a hidden interaction is strongly hinted by flavour).
In some more recent games, roles have been seen where players are self-aware of the Hidden modifier even though they don't know what the Hidden role actually is, with a statement along lines of "There are also additional aspects to your role; you don't know what they are". Even in this case, the word "Hidden" is still generally omitted from the name of the role.
Use & Balance
In some Open games such as C9++, hidden passive abilities are used to preserve balance without making the game susceptible to massclaim; because the game is Open, all players are aware that a hidden ability might exist, and of what possible natures it could have. Otherwise, hidden abilities are usually not seen outside unique roles or bastard mod games.
When playing a flavour-heavy Theme game, check your role's flavour for clues; if you have Hidden abilities, they're likely to be clued somehow.