Short answers, pulled from the story.
A remake rebuilds a game with new assets, engine, and often updated gameplay mechanics, while a remaster updates an existing game's visuals without substantially changing gameplay. The Legend of Zelda: Ocarina of Time 3D is a remake because it uses new character models and texture packs, whereas The Legend of Zelda: Wind Waker HD is a remaster because it retains the original visual style in an upscaled form.
Atari's 1980 release of Space Invaders for the Atari 2600 was the first officially licensed home console conversion of an arcade title. The game became the first killer app for a video game console by quadrupling the system's sales.
Skywind is a fan remake of Morrowind (2002) built inside Bethesda's Creation Engine, which powers Skyrim (2011). Bethesda Softworks approved the project. By 2014, over 70 volunteers had completed half of the game's environment, more than 10,000 new dialogue lines, and three hours of original music.
A demake is a game that adapts a modern title to the technical constraints of older hardware, the reverse of a traditional remake. Examples include D-Pad Hero, a NES demake of Guitar Hero, and Halo 2600, which brings the Halo series to the Atari 2600.
The 2009 remake of The Secret of Monkey Island allowed players to toggle between the original and remade versions at any moment with a single button press. Halo: Combat Evolved Anniversary and Halo 2 Anniversary, included in Halo: The Master Chief Collection, later carried the same feature.
Sega launched the Sega Ages line for PlayStation 2 in 2003, initially as a series of modernized remakes of classic games before expanding to emulated compilations. The series concluded with a remake of Fantasy Zone II that ran via emulation on hardware from the original game's era, making it one of the few enhanced remakes that deliberately avoided modernization.