Remaster - This is the most common category and what most people are usually referring to. Also sometimes called the "HD" version of a game. This is a port of an existing game to a (usually) newer platform which may or may not have improvements made to its presentation. Improvements can be as simple as having a higher resolution/framerate, or as complex as new features, tweaked assets, new lighting effects, etc.
Examples: The Last of Us Remastered (duh), Skyrim: Special Edition, GTA V (PS4/XBO), Wind Waker HD, Devil May Cry Collection.
Remake - Arguably the most desirable category. For the most part, a remake is a brand new game that closely adheres to the design and narrative of a previous game, only now created with modern tools (like a modern engine) and techniques. If not entirely created from scratch, generally speaking, the game will contain more new code/assets than old recycled code/assets.
Examples: Shadow of the Colossus PS4, Crash Bandicoot N. Sane Trilogy, Tomb Raider Anniversary, Resident Evil (Gamecube).
Reboot - A brand new game, with new design concepts, that shares only loose ties to the core design concept or existing characters/lore of an ongoing series. Otherwise viewed as a sequel, or prequel, that throws away most of what has been previously established in an interconnected series of games. Narratively and thematically, it often returns to the origin of the series, and sometimes re-uses the same name as the first in a series, though not always.
Examples. DmC, Tomb Raider (2013), DOOM (2016), Mortal Kombat (2011).
Reimagining - Similar to a remake, only with far more liberties taken with respect to the design and narrative. In extreme cases, it can have attributes of both a remake and a reboot.
Examples: Ratchet and Clank (2016), XCOM: Enemy Unknown, Metroid: Samus Returns, Final Fantasy VII Remake (name is misleading imo).
To be clear, a lot of this can be subjective, and there are definitely many games that fall into grey areas. For example, Mortal Kombat (2011) is a reimagining of the narrative from the first 3 MK games, while from a design perspective, it is a reboot. As another example, Resident Evil (PS3/PS4) can be considered a remaster of Resident Evil (Gamecube) which can be considered a remake of Resident Evil (PS1).