There are several key factors that contribute for a great board game.
* Replayability - no game should ever be the same, otherwise you'll get bored; this is usually solved with different starting locations, different starting powers, randomly generated board, etc.
* Interesting theme - makes the game stand out in a crowd.
* Exciting rules - because emotions is what the game is about. The best game rules could be described as "fast, fun and furious". Perhaps omitting the furious part.
* Nice look - it's not THAT important, but a ugly looking game scares away people; so called "American style" games are famous for their great looks (but sometimes criticized for uninteresting gameplay).
* Balanced game mechanics - so the gameplay seems fair for all the players, and there is no simple winning strategy.
* Good level of interaction - this is a hard one; basically, you'll find that "European/German style" games have a lower rate of interaction between players, which can be a issue for some gamers. On the other hand, not everyone likes games based around conflict and fighting with other players. Make your own pick.
* Easy to learn, hard to master - one of the best games in the world are based on this simple principle. The more complicated the rules, the more probable you'll get bored before finishing the manual. Not to mention all the mix-ups and bookkeeping during play.
* Number of players - a good game has a mechanic for scaling the game according to the number of players. Some games omit this concept and tend to work good only with a certain number of players
* Playing time - this is also an important factor. Short, party games (sometimes called "filler games") usually take from 15 to 45 minutes. The best playing time for bigger games is from 45 to 120 minutes. Of course, there are games that can be played for 6-8 hours, but not everybody has the time and desire to spend half a day playing board games.
Also, there are some things you'd like to avoid in a board game:
* Snowball effect - when the leading players earn even more points/resources that allow them to win easily
* Kingmaking - when a loosing player can decide over a tie between wining players
* Analysis paralysis - when the game is overcomplicated and it takes a long time to figure out which move to take
As for my favorite board games: Power Grid, Attack!, Citadels, and Starcraft: the Boardgame.