I used to be baffled by their continued existence. Then I found out that all board games weren't Monopoly.
I really like the genre known as German board games; they tend to emphasize skill-based gameplay using a variety of interesting mechanics over anything else.
Good ones tend to lead to a lot more balanced and interesting decisions than most video games do, actually. They can't use reflexes and don't have the graphics race to be concerned about, so they often tend to focus on innovative gameplay mechanics and balance over more cosmetic aspects.
The day before yesterday, I just played a game of the new Pandemic expansion and it was pretty good. Puerto Rico, Power Grid, Small World, and Race for the Galaxy (though technically a card game) are all great games that I have and would recommend. Other great ones are Pandemic, Arkham Horror, Cosmic Encounter + Betrayal at House on the Hill (though they're not nearly as balanced as the others), Battlelore, Dominion (technically a card game), and Agricola.
If anyone is interested in these types of games, I'd suggest looking at Boardgamegeek.com. It's a hotbed for really good board games.