This is my favorite new board game:



It was made in the spirit of heroes quest, but it is for two players, and the advantage isn't always for one side or the other.

A classic Fantasy Boardgame is "Talisman".
http://store.fantasyflightgames.com/showproducts.cfm?Step=1&FullCat=118

Wasabi! and Bohnanza are good games that can be played with kids and the rest of the family.

I will second the recommendation of Wasabi, Ticket to Ride, Carcassone, Formula D, and Lost Cities.  

I will also suggest looking at Small World (a game where you play varying fantasy races in order to gain the most victory points in a short amount of turns...I think it's 10 or 12 turns), Agricola (a game about farming and managing responsibilities with limited people...a Euro game), Pandemic (where you work WITH the other players in an attempt to stop the spread of diseases across the world), The Adventurers (a game where you play an adventurer going through the ancient ruins, dealing with collapsing walls, a rolling boulder, lava pits, bottomless caverns and swift rapids....collect artifacts and make it out alive!)

A couple of notes on the games I listed first.  Ticket to Ride has a variety of different versions.  The "American" version is probably the easiest to learn on and can handle the most people if you want others to play.  The Nordic countries and Switzerland are more limited on players that can play, and they include more different things that take a bit to get used to.  So you might want to learn on the American rail system.
Formula D is the reprint and revamp of Formula De.  It's a lot of fun and allows for normal racing and "street racing" so it has a couple of options.  Plus, I know you can find some maps online (a friend of mine has done this) and print them out for new maps to use.
Lost Cities actually has two versions; the card game and a board game.  The card game is strictly cards and is only two player.  The board game can handle up 4 (maybe more, I can't remember) but it does have pieces to use (it uses meeples and thick, large cardboard pieces, so its not too bad).  

For younger kids there's games like Hey, that's my fish! and Lemmings.  For kids of about age 7+, depending on the child, there's Hive, which is a game that's like a mix of dominoes and chess.  The heavy duty hex tiles have the feel of dominoes and have types of bugs on them.  Each bug does a different thing.  You try to get to the other person's bee to win the game.

Hope this helps some.

If you want to do WW2, I'd recommend Conflict of Heroes. It's a tactical game, but pretty simple compared to stuff like ASL. Conflict of Heroes

If that's too difficult, Axis and Allies is great for kids.

I know it's not a boardgame per se but I ran rpgkids for my two kids a few weeks ago and they love it. The adventure that is included in it has a cartoony gameboard you can print out. Instead of the cartoon character 2d game peices I let them pick out one of my minis they felt best represented their character. They loved it though they both told me later they wanted to level up their characters and get more skills. Next one the block is The Warriors which is actually free off their site. My son is not too into the whole roleplaying side of it so I keep them playing on a gameboard as much as possible.

Played Last Night On Earth last night (zombie boardgame).

I do not recommend it.

Gameplay seems too random: it's basically a combination of luck of the dice and mining the deck for things like weapons/first aid.

For instance: to resolve a fight the zombie player (who controls all the zombies) rolls 1d6 while the hero player (there can be more than 1) rolls 2. Zombies win on ties (hero takes a wound), while highest result wins - but the zombie only dies if the hero rolls doubles as well!

And to gain cards the hero forfeits his movement to draw a random (ie potentially useless) card. Cards are vital.

Urgh.

I also played Last Night on Earth. It's weird. Heavily luck based, but it's for both sides (Zombies and Survivors). I wouldn't buy it as when I'm looking for that fiddly of a board game, I already have Arkham Horror.

A game I just played tonight and highly recommend is Betrayal at the House on the Hill. Setting up the game is part of the game itself, and then the game actually happens.

To explain WTF that means:

The first act of the game is exploring the Mansion. Players place down card as the explore a la Zombies!!!! All players generally want to work together at this point. Some events happen, some characters get buffer (or weaker), you find some things when suddenly...BETRAYAL!

The second act comes when someone triggers a betrayal. The circumstances of the last turn determine exactly what kind of Betrayal occurs. The entire objective changes at this point. Bust out the secret rulebooks and it's time to change the game. One player (determined by the specific betrayal that occurs) becomes the traitor and the others have to stop them/survive/escape etc.

The betrayal in the game I played had a Lovecraftian plague seeped in to the house and begins to kill everyone slowly. One random player (determined by a secret draw) was corrupted and needed to corrupt or kill everyone else. The other players had to scramble between the lab and the greenhouse to make a vaccine and vaccinate all non corrupted players. If everyone is corrupted, the last uncorrupted player loses. If all remaining uncorrupted survivors are vaccinated, the corrupt lose.

This was one of 50 endings. 50. I'd heard the owner of the game talking about his last game which had the traitor as a servent of Dracula, who was trying to wake his master. Pretty varied stuff there.

I'm going to be purchasing this game at the next possible opportunity. It gets a rec from me

LoE is thematically interesting. But the problem with it's luck based mechanism is that it promotes boring gameplay: in order to draw cards you must forfeit your movement (which you can do after rolling to see how far you can move and only inside a building). This of course leaves you completely vulnerable, but worse it means you don't do much and can only hope you draw something useful. Most of the cards are fairly bland with a few very powerful weapons (chainsaw). Given how hard it is to take down a zombie, rather than just survive the combat (you are considered to have fended them off) it becomes tedious. A shame really.

I really would like to try Mansions of Madness.