Even/Odd Night is a role modifier that enables a power role to act on either Odd Nights or Even Nights only. Typically this modifier is attached to roles as a method of weakening them, reducing the amount of times in a game they can use an action.
Use and Power
Although an Even/Odd Night role might at first seem half as powerful as the full equivalent, this is normally not the case in practice. Depending on the size of the game, some nights are often more important than others, e.g. it's more useful to have a Cop investigation result on the night before LyLo rather than the night after; additionally, a common risk with power roles is dying before they get an opportunity to use their abilities, and this is somewhat more likely to happen with an Even Night role than with an Odd Night role. As such, especially in smaller games, choosing a role as Even Night or as Odd Night can be a good way to fine-tune the balance of a setup.
Some moderators like to use Even Night and Odd Night roles in pairs as a method of reducing swing (because a power role dying can cause a lot of swing, and splitting the ability across multiple players reduces that), but it's sufficiently common to see the modifier on its own that players should not assume that the other modifier also exists. Note that (at least for Town) a split pair like this is more powerful than the role would be on its own, because it gives more scope for players' claims to be confirmed via their actions, and because Mafia may successfully identify a power role (using, e.g., a Tracker) but then attempt to roleblock them uselessly on a night where they couldn't act anyway.
When multiple scumteams exist in a setup that's too large to have multiple kills, it's very common for one scumteam to have an Even Night kill and the other to have an Odd Night kill. This is also typically the case in a setup where a single scumteam is split into multiple separate "mini-teams" that win together but can't communicate with each other.