Although "Weight" isn't precisely defined, the game pages now reference a "complexity rating" which they define as "Community rating for how difficult a game is to understand. Lower rating (lighter weight) means easier." (Source: the ? next to Weight on game pages)
For different people weight means different things, usually a combination of things like:
- How complex/thick is the rulebook?
- How long does it take to play?
- What proportion of time is spent thinking and planning instead of resolving actions?
- How hard and long do you have to think to improve your chance of winning?
- How little luck is in the game?
- How much technical skill (math, reading ahead moves, etc) is necessary?
- How long does it take to learn the rules?
- How many times do you need to play before you feel like you "get" the game?
BGG uses a 5-point Weight scale:
- Medium Light
- Medium Heavy
Weight is part of the rating system.