Magic Maze is great for newbies - rules are simple and introduced over multiple scenarios. I've had success with both kids and adults.

If you like word games, Illiterati is a cooperative word game. It plays a bit like Bananagrams in that you're having to accommodate additional letters into your sets of words, and each player also has their own category of words they're trying to complete.

Red November, Forbidden Island, and Forbidden Desert are all pretty similar "Move your pawn around and do things before everything blows up" type games similar to Pandemic.

For beginner friendly co-op games I'd recommend:

• The Crew which is a trick taking card game, and has a great legacy/mission system, easy to learn
• Forbidden Island
• Burgle Bros

If you're looking for something geared more introductory to co-op board games, Castle Panic is good:

https://boardgamegeek.com/boardgame/337394/castle-panic-deluxe-collection

I prefer co-op games over others, so I love this question.

• Castle Panic
• Pandemic (with the easy difficulty level for beginners)
• Eila and Something Shiny

Did no one mention LotR: Journeys in Middle-Earth? Its a great coop game, the only con being it requires an app to play. There are few expansions that introduce more playable characters and campaigns, with standalone campaigns being purchasable via the app.

There are lots and lots of co-op games. The former top 1 on bgg, Gloomhaven, is a co-op dungeon crawling rpg in a box.

If you prefer kicking some eldritch asses together there are all the Eldritch Horror, Arkham Horror, Mansions of Madness and Elder Sign, where you and your loved ones can go (globe/city/mansion)-trotting to beat some abominations and save the day. And if you like this cthulhu theme and card games, there is the wonderful Arkham Horror CCG which is basically the same premise, but now with deck-building and more in-depth stories.

Another incredible one is Dead of Winter where you and your friends simulate being the last humans in a colony surrounded by zombies. It can be full co-op, but the best thing of DoW is it's hidden traitor mechanic, where someone can be working for their own goals instead of the colony.

If you like simple rules and lots of surprises I recommend Betrayal at House on the Hill which starts co-op and after the haunt of the house is revealed anything can happen. The game starts simple and ok and goes on a crescendo of twists and turns that leaves you wanting more.

And finally if you just want to sit around with people on the table and solve something together like a puzzle, I lived Sherlock Holmes Consulting Detective. A game where you and your fellows are the street boys of Baker Street helping (and in this case competing against) Sherlock to solve mysteries. It has basically no rules and everyone work together to solve the mystery as fast as possible.

Coop games are tricky because many allow “quarterbacking”, which, while really being a player problem, I blame on bad game design. (For example Pandemic could really be played solo with 4 stations)

So you need coop with asymmetrical information and nobody can have all the information.

Hanabi is a great classic example of a game where nobody can see their own cards, but there is a limit to the amount of info you can give out.

Just One is a newer example of a coop party game.

If I may add some less well known (but still fun) cooperative games.

Insider (Oink Games) - 20 questions with a twist. The twist being that one player knows the secret word and wants people to guess it, without being found out.

Hey Yo (also Oink Games) - small and cute rhythm game where we try to place cards on the beat, scoring points for uninterrupted lines of the same color cards.

Rail Pass (Mercury Games) - play this with people who are turned off by overtly competitive train enthusiasts. You try to hand over trains filled with cargo without dropping it and getting the right cargo to the right player. But you have to make train sounds before handing it over to them. Great fun. Especially with adults.

Romeo & Juliet (Sylex) - a challenging two-player game of unspoken communication, where you try to get Romeo & Julia to fall in love with each other while still making it to the end alive.

Kreus (CMON) - similar to the above, but for up to 4 players. Create a planet by playing cards in the right order using only logical deduction and sparse information. A brainteaser that feels amazing when you pull it off, but need to pay close attention and think two turns ahead to do so.

Space Cadets (Stronghold Games) - an underrated Star Trek style game of wild, real-time cooperation. Players each have individual roles (and rules) to fulfill at their station. Only if everyone pulls their weight do you stand a chance. It's not a very elegant design, but once you make it past the learning curve it's a fantastic experience.

It's been mentioned, but Spirit Island is an excellent co-op game and the Horizons of Spirit Island box is intended to be more of an introduction set for the game with lower complexity.

I really enjoy Forbidden Island for a fairly quick co-op game which isn't too complicated.

There are some great suggestions here already. I'll throw out some of my favorites that are all relatively inexpensive, easy to find, and easy to learn:

The Mind
Incredibly easy to learn and teach. I don't think I've ever played just once game because everyone always wants to play again immediately.

The Crew
Great cooperative trick taking game. There's a reason it won a bunch of awards, and the sequel is even better. Can't go wrong with either. Regular filler game for our game nights.

Just One
A party game where you have to guess a word. Everyone is the clue giver except one person (rotates every word) with the catch that all clues are one word, written in secret, and if two are the same then they get hidden and can't be used. We have gotten a ton of mileage out of this one. Lots of fun, 10 second teach, good for a group of 5+

Gloomhaven: Jaws of the Lion
It's admittedly a little complex, the only game on my list I wouldn't describe as easy to learn (unless you love tactical rpgs, in which case it's not bad), but there are great online tutorials you can watch on YouTube. It has a video game feel. You're 2-4 heroes fighting against the bad guys I a combat map that comes in book form. You can find consistently for $25 or less in the US, sales just keep popping up. Without question the best starting point for the Gloomhaven universe.

A few that are relatively easy to learn:

• The Crew (either one)
• Forbidden Island/Desert (I prefer Desert)
• Pandemic or Flash Point
• Micro Macro
• One of the Zombicide games
• Lord of the Rings: Journeys in Middle-Earth

Increasing in complexity or length:

• Burgle Bros. 2
• Spirit Island
• Arkham Horror 3rd Edition
• Kingdom Rush

To name a few!

I'll echo some suggestions and add some more. Before that, there is something to note about many coops; you can divide them into two groups, first, could I play by myself (e.g. Pandemic and controlling multiple characters), and second, that they require multiple people because of hidden information (e.g. Hanabi). Almost all of my coop suggestions fall into the later.

Just One - @roarmalf pointed this one out, and I second it. This has gotten an extensive amount of play in our group.
Quirky Circuits - This is a real time game for the action allocation portion where you play your cards. More players is better IMHO. You're playing cards to program the little roomba/robot that is cleaning the house and then you flip them up to see where it actually goes.
Rosetta: The Lost Language - I think it's a marriage of Mysterium (coop) and First Contact (competitive) as a small box coop. Incredibly group dependent, but there is a print and play copy (of the first edition) available for $5 from a different publisher who is going to do a second edition in the future. this one scales well once you get 3 total people as a bottom threshold and players can enter and drop out of the game (other than the story teller) at will which is a nice feature. Some people will like this more than Mysterium (and it has 1/4th of the rules), some will like Mysterium more.
Magic Maze - Kasper Lapp has a new one called Gardeners that I really like, but it's not available widely yet where as his earlier game that is quite similar called Magic Maze is. Players can't communicate except by tapping a wooden peg in front of another player to alert them that they should be the one doing the next move. otherwise it's unstructured as to who should be taking the next turn (oh, and it's real time).

My wife and I mostly play cooperative games, she doesn't like very complicated games so we've been through a few of them:

Pandemic - great base game for cooperative but does have an issue with quarterback and no way to mitigate so you need to be cognizant of this. If you really enjoy the theme/playstyle of this I hugely recommend getting the legacy version (Season 1 is the best but I've heard good things about the others)

Marvel Champions - This is A LCG so can be a steep price if you want literally everything, but the base game itself is fun enough so I'd suggest starting with just that, and if yall enjoy grab either the expansion that sounds the most interesting or the boosters for whatever hero one of you enjoys the most. You have rules for building custom decks but all of the heros come with a pre-built that works reasonably well with their theme

Gloomhaven - we've found that my wife is much more willing to take on a more complicated ruleset as long as it's a legacy type game where the base rules stay the same, and you build towards something. I haven't personally played it but a lot of people recommend starting with Jaws of the Lion first if your new to the genre as it's cheaper and has a simplified ruleset

One Deck Dungeon - simple ruleset dungeon crawler that takes up little shelf or table space. Great little game, note if you want to play with more than 2 you will need the forest of shadows expansion.

Mansions of Madness - The second editions app works quite well at reducing the complexity of the game while keeping it interesting. We'll play this often on Halloween with friends while waiting on trick or treaters.

Call to Adventure and its variants are played co-op.

Horrified is a great co op board game. Can mix it up a few times.

Replayability will be limited, but I've only played it around 4 times since owning it October 22.

Lots of great games mentioned. If you have younger ones that you want to bring in to the fold, there’s a couple good ones I’ve had success with:

• Sky Magic: you play as fantastical creatures trying to brave an oncoming storm, everyone needs to get to safety to win, so some strategizing together to optimize each other’s paths is rewarded.
• My first Castile panic: simplified version of the classic suitable for the younglings.

Library Labyrinth recent delivered its kickstarter. The basic premise is that a bunch of evils have escaped from the books in a library, and you have to use great women from literature and history to defeat them and shove them back in their books. The first time I played it I got to say the immortal line "Mina Harker, in the library, with a photocopier!" It's a lot of fun and pretty quick to pick up, and the production values are very high. Definitely a recommend from me

Stardew Valley has a coop mode, in essence the game is about starting a farm but it's much bigger and more entertaining than it sounds, seriously worth a try.

I'd recommend Sentinels of the Multiverse. It's a fun co-op card game about superheroes trying to defeat a supervillain. It can get a little expensive if you try to buy every set in the series (my friends and I have done exactly that and it's a massive number of cards), but all the different sets are compatible with each other, so you can pick the ones you want and just get those. The core gameplay loop is pretty simple where the villain and the players all take their turns attacking, but there is a ton of variety in how all the characters can be played.

https://boardgamegeek.com/boardgame/102652/sentinels-multiverse

Cooperative games are a bit rare I think. But my family has "Pandemic" which is pretty fun.