In some ways, it's unfair to claim to love Samurai Swords because I've played it fewer than a dozen times in my life.
I tend to think of board games (really meaning all tabletop games as distinct from video and computer games) sharing a few broad characteristics.
3) Strategy v. Luck
4) Interaction with other players
5) Zero-sum v. Cumulative score
My all time favorite games, including chess, Settlers of Catan, and Dominion all feature oodles of all four. Chess, my favorite pasttime, is both intensely complex and entirely dependent on interaction with the other player. It is one of the few truly complex games with no luck involved. The thing I love about chess is that if you win or if you lose, the pride or the disappointment are entirely yours to own.
The other two games I've mentioned depend on some luck; Catan is played with two six sided dice, while Dominion is a card game so the order after a shuffle determines everything.
I'm a fan of less luck, more skill games. I also, generally, prefer cumulative score games like Catan (where play continues until someone reaches 10 victory points) over zero-sum games like Monopoly where play continues until one player has taken all resources away from the other players.
I'm currently addicted to an old and wonderful board game called Samurai Swords (alternatively sold as Shogun or Ikusa, the owners have known they have a great game but the marketing obviously escaped them). It's a Milton Bradley Gamemaster game, so it's layout and basics compare to Risk and to Axis and Allies.
But Samurai Swords is to Risk as calculus is to the multiplication table. Instead of receiving troops for ownership of land, the player receives money ("koku") that he can then spend on different kinds of troops (1 koku buys a bowman, or 2 samurai, or 2 gunners, or 3 spearmen), on mercenary ronin, on the ninja assassin, on building fortresses, or on choosing when in the player order he wants to take his turn.
The downside I'm struggling with right now is the luck involved in victory. Each of the different units has a different chance of scoring a "hit" on a twelve sided die. While each player builds his army, and thereby determines it's quality, critical battles can turn on a couple lucky rolls. Unbelievable last stands are not as common as in Risk (the defender has no die-based advantage in Samurai Swords), but ambition thwarting surprises still abound.
How many rolls of a twelve sided die are necessary to achieve a significant sample size? How many battles would have to occur from a moment of resource equality in order to be fairly confident that player skill, not luck, have determined the outcome? When has a player crossed the threshold of defeat, the point at which the odds of mounting a comeback become impossible given his current resources?