Some of these fall into multiple categories, and many are Xbox Live Arcade games (noted as XBLA). In fact, XBLA probably has the best selection of these games, and you can buy them right from your 360.
Braid (XBLA)
Lumines (XBLA)
Limbo (XBLA) (violent in an implied way, but also very... zen?)
Chime (XBLA)
Lego Games, especially Harry Potter and Star Wars - and there's even a Lego Rock Band to fit your desire for music games
Texas Heat Poker (XBLA)
Almost anything Kinect-driven (if you don't have one, get one - it's fantastic for casual games), including:
Kinect Sports
Kinectimals for cutesy stuff
Child of Eden
Your Shape Fitness (AAA title)
Portal (XBLA) great puzzle game
Portal 2 even better puzzle game
Feeding Frenzy (XBLA)
Tower Bloxx Deluxe (XBLA)
ilomilo (XBLA)
E4 (XBLA) (rhythm-based game)
Peggle (XBLA)
the Fable series. There is fighting, but it's not excessively gory, and is pretty casual for an RPG.
Catan (XBLA)
Carcassone (XBLA) these are two German board games translated to the 360, and are quite fun. They both fall into that easy to learn, difficult to master kind of thing.
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