First impressions set the tone for a relationship of any kind. First impressions in the board game realm are just as important as impressions with people. Monopoly, Risk, Life, Trivial Pursuit, and games in that ilk are the standard of boardgames to most people. The bar is yet relatively low, but many people have a comfort with these games, and that has shaped what games should be in their mind.
Because of these preconceived notions, I feel like every gamer needs to have a game that they can pull out to show people what a designer board game is. Now, I run the risk of sounding really snobby here, so I'm doing my best to dance around that. Ultimately, we play games to have fun. If people have a good time playing Scrabble or Clue or Monopoly, that's great. I've done my best to never discourage people playing games, no matter what the game.
The goal of introductory games is to show people that there is a whole world of possibilities in games that they may not know about, and those games are something they might enjoy. As always, I'll provide a few examples, give my thoughts on them, and hopefully help you reach an idea of what you'd like to have in your collection.
Settlers of Catan (3-4 players in the base, 5-6 with expansion)
This was the game that started it all for me. It was my first exposure to a game where everyone played the whole time, and had things to do even when it wasn't their turn. If you're not familiar with Settlers, you are trying to build settlements and cities with resources that you get on dice rolls. You'll rarely have everything you need, so you have to trade with other players. The dice are a comforting part of the game to players. The idea of trading is familiar, but the frequency is new to people. A lot of people can really get in to Catan and have fun the first time. Catan appeals best to the dice fans, and people who like Monopoly.
Carcassonne (2-5 players, 6 with Inns and Cathedrals expansion)
Dominion (2-4 players, up to 6 with Intrigue)
I first played Dominion in 2009, and I was instantly hooked. I bought it for my birthday, and it started a bit of an obsession. I've always been a fan of card games, I dabbled in Star Trek CCG and Pokemon, so I liked seeing cards interact. The idea of deckbuilding always intrigued me, but as a kid, I never had the money to have the best cards or a lot of the good cards, so I knew I'd never be great. Dominion changes that idea. It's deckbuilding, but the whole game is building your deck. Everyone starts with the same deck of 10 cards - 3 victory points, and 7 money. You have to use these cards to buy better cards, slowly making your deck bigger and better. The base game comes with 25 unique kingdom cards, and you play with 10 in each game. This gives a lot of variety and makes most games feel different than the last one you played. If someone is a fan of a CCG like Magic the Gathering, or Yu-Gi-Oh! this is something for them to look into. They might just appreciate not having to spend $100+ every few months to get new cards.
Ticket to Ride (2-5 players)
Ticket to Ride is the first game I ever bought for myself. I played the German board (Marklin version) with my gaming family. That's the most complicated version of TtR, but I liked the core. I decided to get the US map, since it would involve the least amount of hunting for odd cities, and thus save time. I'm so glad I bought TtR. It's currently my 3rd most played game (if you count all the different maps, 4th if you don't) and I don't see that changing. It's a game I can play with just about anyone. On a player's turn they have 3 options, really only 2 that come up every time. You either take cards or play cards. The third option is taking more destination tickets, but that isn't something to do every turn. The scoring is fantastically simple, you get points for trains on the board, you gain points for completed tickets, you lose points for tickets you don't complete. It's hard to nail down a particular group of people that would be best suited for the game. I don't want to cop-out and say everyone, so instead I think that Ticket is a great way to get people who used to play games as a kid, but lost interest due to a plethora of things, back into gaming.
No, not a game, just time to talk money.
Like all these categories, this comes down to the budget pick, and the pick you want to have around.
Carcassonne takes the budget pick, but if you're willing to go up a bit, get Dominion.
As for the game to have around, my winner is still Ticket to Ride. It works for the widest range of people, and trains are just fun. If you want to take a 2nd in any category, I'd encourage you to take it here and also get Dominion.
These 4 games are fairly common these days. I think Target even carries at least Ticket and Catan, maybe more, haven't checked lately. Since they are common, you can generally find someone who owns a copy of at least one of these and ask to try it out. I like all 4 of these games, you can't go wrong with any of them.