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?
Etc...
nuff said