Oh there are so many; it'd be great to know what kinds of thigns you're into, but here's a generic list of modern classics that are tried and true:

Carcassonne - Really simple and elegant tile-laying game that is full of strategic decisions for players of all levels. (If you like the tile-laying, continue on to Castles of Mad King Ludwig or Suburbia)

Dominion - Introduced the "deck-building" mechanic where you buy cards for your own personal deck over the course of the game and manage probabilities of drawing what you need (this later blossomed into a HUGE genre; good next steps are Valley of the Kings, Arctic Scavengers, or Star Realms).

Pandemic - Cooperative game about battling disease outbreaks across the globe. In some ways it can be easier for new gamers because you're all working together to solve the challenge. Spawned many expansions, as well as the aforementioned "Legacy" version, which is effectively a campaign mode with permanent changes game-to-game. (Also check out Flash Point: Fire Rescue, and Forbidden Desert/Forbidden Island.)

Coup/The Resistance - These are social deduction/bluffing games where players have hidden roles. They offer slightly different experiences: Coup is slightly more strategic and about selecting the right roles to come out ahead, while The Resistance is a little simpler but brings out more tension (because it's team v team). (Also check out Avalon a Resistance spin-off. If you want a little more tactical structure behind the deduction, checkout Citadels)

Hanabi - A really unique cooperative game about communicating information effectively. You can see everyone's cards but your own, and as a team you are trying to lay cards down in a specific order, but you are limited in the kind of clues you can give to others, and clues are a limited resource. (There isn't much that's super-similar to Hanabi, but a more advanced and thematic spiritual successor is coming out in a few weeks called Beyond Baker Street)

Sushi Go - Elegant little "card drafting" game where you take a card from your hand and then pass the hand to the next player, trying to collect sets of sushi to earn points. Despite its simple mechanic, it's balanced, quick, and fun for all types of gamers. (In this genre, I'd really recommend 7 Wonders, which is based on the exact same mechanic, but involves building an ancient civilization and brings rich resource management challenges into the game. One of my favorite games to this day; you could even start with this, but it's definitely a higher complexity level and it would be very helpful to have someone teach.)

Ticket to Ride - Very straightforward game about collecting sets of cards and using them to place trains on a map. One of the few games I can get my dad to play that we all find fun -- new players can get into the game quickly. Spawned a ton of spin-offs, so it's definitely loved.

Lords of Waterdeep - Great introduction to the "worker placement" genre. Players move tokens to take actions, as well as preventing others from taking the same action. The actions let you collect followers, construct buildings, and send them on quests (the game takes place within an official D&D setting). I don't personally love this one because I got into more advanced worker-placement games before I played it, but I can understand why it's adored by so many people. (Interesting next steps: Viticulture or if you want to jump off the deep end Caverna/Agricola, which are punishing but both have spent time rated as BGG's highest ranked game)

If that's not enough to look up, here are some other games I don't have time to describe, but I'd also recommend for newbies: Splendor, Takenoko, Ra, FUSE, Survive: Escape from Atlantis, Bohnanza

And some more excellent "next steps" that are a bit higher in complexity: Power Grid, San Juan, Dead of Winter, Cosmic Encounter

(And in all this I hardly listed any party or dexterity games, so if you're looking for those, check out the resources below.)

Good resources to read/watch more:

- There are a ton of news/gameplay/review sites, but I think Shut Up & Sit Down[1] really takes the cake on having entertaining pieces that explain exactly what's interesting about each game. They also do a lot to earn the title of "ambassadors to board gaming" and target much of their content at newbies. They have a really good video titled "New to Boardgames"[2] that's great at getting people excited about the hobby.

- reddit.com/r/boardgames has been a really positive and helpful community in my experience

- BoardGameGeek[3], which is THE reference point for the board game community. The interface is clunky, but you can't beat the comprehensiveness of the database.

- For other video shows, I'd recommend: "Rahdo Runs Through"[4] for game introductions and overviews, "Watch it Played"[5] for more in-depth tutorials, and Wil Wheaton's "Tabletop"[6] for general "let's play" content.

[1] https://www.shutupandsitdown.com/

[2] https://vimeo.com/70647384

[3] https://boardgamegeek.com/

[4] https://www.youtube.com/user/rahdo

[5] https://www.youtube.com/user/WatchItPlayed

[6] http://geekandsundry.com/shows/tabletop/