The vast majority of board games available today are fun board games for adults. They may be fun for other age groups as well, but they fit in the adult category because adults have already lived through those younger years. It's what qualifies them to be adults.
That said, it should be obvious that not all board games are fun for every adult. Fun board games for adults I'm associated with might not be the same fun board games for adults you have contact with. Among the dozens that I've played as an adult, there have been a small fraction that I haven't particularly enjoyed. They are few and far between though.
When speaking of adult board games, I'm not going to devote time to those that have risque content. The adult-ness referred to here will only be a reference to an age group, not to games that are for "adults only" because they would be rated R if they were movies.
There are some traditional gateway games mentioned fairly often by those who make suggestions for board games for adults - and I think understandably so. Ticket to Ride and sometimes Carcassonne are most frequently brought up, and I agree that they should be as I would also recommend them, especially for adults who have not yet played anything remotely like a Euro board game before. Yes, that's how these fun board games have earned the moniker of gateway games. Once people see how much fun these are, they tend to look for more fun board games that they haven't played before.
Sequence is a combination card and board game for 2-4 players. It is somewhat reminiscent of the classic Pente in that you need to get a row (or two, depending on the number of players) of five chips - vertically, horizontally, or diagonally - for a victory.
The board shows all the cards in the deck twice except the jacks, which are wild and have special properties. Play a card and place a "poker" chip of your color on a corresponding card space. Then don't forget to draw a card to replenish your hand. If you don't pick up a new card before the next player plays, you have to finish the game with less cards in your hand than you deserve, which can severely limit your options - especially if you forget more than once.
Sequence is a light-weight board game strategy-wise, so there's plenty of time for socializing, if that's what you like to do while playing with friends and family. Still, there is enough challenge to make you think a little before placing just any old card anywhere.
Wits & Wagers
As its name suggests, Wits & Wagers is a combination trivia and betting game that is great for parties. It plays quickly, especially if you enforce the time limit imposed by the hourglass. And it doesn't really matter if you know the answers to the trivia questions; you can still win by intelligent wagering.
A question is read from one of the cards included. All answers are numeric. (This includes years.) Players write down their answers secretly within 30 seconds. When all are ready, answers are revealed simultaneously and are arranged on the betting table from low to high. When sorted properly, players then have 30 more seconds to place their "poker" chips on the answer they think is closest (without going over) to the correct one.
Winnings are paid only to the player or players who guessed the correct (closest) answer according to the odds shown on the betting table. The farther away from the median answer, the higher the odds.
Alhambra is another combination card and board game. Cards are money in four flavors usually identified by card color (though other cues can be used by those who have difficulty with colors) and come in various denominations. Normally, you pick up cards on your turn or discard some of your cards to pay for one of four tiles available on the board.
These tiles, most of which have dark-colored "walls", are used to construct your own personal Alhambra. The tiles also come in different colors. Twice during the game and once at the end, players score points based on how many tiles of each color are in their Alhambra.
The challenges come in deciding when to purchase a tile, when to bide your time and collect more cash, where to place a purchased tile in your Alhambra (as there are restrictions), and which tile to go after.
When playing with three or four people, you can usually do a fair amount of planning ahead. With five or six people, chances are that the tile you were waiting to buy is going to be taken by someone else before your next turn.
With just a little more strategy involved than in Sequence, this board game will provide many hours of fun for most adults.
Party games like Outburst and Balderdash are two more fun board games for adults that you might like to try. Balderdash is virtually the same as the generic "dictionary game". Having the actual game just makes finding good words to stump people with a lot easier.
Outburst is similar to Family Feud in that you're trying to come up with lists of items in a category as a team. The main and significant differences are that everyone on the team participates at the same time and that the items in the unknown list are not necessarily the ten most popular answers. Sometimes you'll wonder why in the world they didn't list an item that your team came up with, and other times you'll be puzzled at where the creators came up with an item that none of you thought of.
Quiddler is a card game that works best with players of roughly equal vocabulary levels and spelling ability. You have a hand of three to ten cards (depending on which round it is) that have letters instead of numbers. After drawing a card, you try to spell one or more words using all the letters except one, which you will discard. When one player accomplishes this, the rest have one more chance to do the same, or to at least play as many cards as they can. Any cards left in your hand score negative points. Bonuses are awarded for the longest word and most words each round.
With these suggestions as a starting point, you should be able to find your way to many, many other fun board games for adults.