A good remaster is one that upgrades the existent media to modern platforms without remaking its contents entirely.
So;
-Enhancements in texture and lighting
-Improvement on controls and options around controls customization
-Widescreen support if applicable
-Higher Resolution output
-Improvement in sound and music (Sometimes just an uncompressed version of the original audio is enough)
-Improvements and possible additions of cut or planned content that was left in the cutting board floor, now finished and reintroduced
-QoL improvements like Quicksaves, Rewind, Fastforward (Useful for older JRPGs)
-Balancing and bug fixes
Good examples of remasters imo, some definitely border on the Remake territory. but their fidelity to the original is close enough to list ;
-Metroid Prime Remastered - Honestly, stellar accomplishment, especially for the hardware
-Resident Evil HD Remaster - It is a remake, but one that stays very faithful to the original, with improvements and additions
-Dead Space Remake - It is a full remake, but unlike RE2,3 and 4 that are full remakes with some reimagining, the fidelity of Dead Space to the original is pretty good, and thus worth mentioning
-Final Fantasy XII - The Zodiac Age - A fantastic update to the original PS2 release, with many improvements
-Black Mesa - Again, it is a full blown remake, but it is to me the definitive way to play Half-Life now
-Star Ocean The Second Story R - This one is a bit on the middle of the road, it is a remake, but also a remaster, and an all in improvement package. Fantastic way to enjoy the game
-Final Fantasy Tactics - The War of the Lions - More of an improved version than a remaster, but a fantastic one nonetheless
-Grim Fandango Remastered - To the point! As is the other Double Fine Remasters of LucasArts games
As a bonus the upcoming The Thing Remastered - What Nightdive has shown is delicious, can't wait for it!