I will build on filoraene's writeup with as many details as I can provide.
Agricola recently turned 15 years old, and they're making a big "Agricola 15" box with nice shaped pieces and several expansions all combined into one box. My own copy is aging, its box worn, its box top's corners split. We long ago used up all of the included scoring sheets. We played it so much that I have come up with a little process to make sure that all of the resource spaces get updated each turn, and speed up the game. I can't emphasize enough how much fun Agricola is! It's great.
Agricola is a "worker placement" board game, a genre originated by Caylus. In those, players have a limited number of worker tokens that they take turns placing. A placed token grants some function or advantage, but once placed other players can't use that same space while the worker is there. It usually goes away, or "returns home," after the round ends and all the workers have been placed. This means the first player to go in a round has an advantage, in that all the options are only available to them, so usually there is some mechanism to change the starting player during the game. In Agricola, this is done by using one of your workers to take the "Starting Player" space, which makes you the starting player for subsequent rounds, with following players going after you clockwise, until another player takes it themselves.
Like in many worker placement games, the players don't directly interact often. Players indirectly interact all the time though, since their workers block options, either accidentally or on purpose, and further players take resources that other players were hoping to collect. Agricola has been designed so that players are constantly having to focus on building up their farms. There's so many things you want to develop and only 14 short rounds to do it in. In Agricola, the end of the game is in sight from the very beginning, it never feels like you have all the actions you want, and it's a relief when you actually manage to use up your whole board. I notice, because of this, that playing specifically to block other people is usually a bad move, because it's one less turn you spend on improving your own farm. By making good moves for yourself, though, you will naturally block many other players. This helps to liven Agricola up, while meaning it's difficult to actually blame other players for messing up your game.
There is nothing you can actually do to reach onto another player's board and take their stuff. Even though there's hundreds of Occupation and Minor Improvement cards you can get, even before including expansions, none of them outright steal the others' food, crops, animals or resources. But in addition to the worker-placement interaction, players can get options that piggyback off of other's players moves, give them advantages that activate when other players go, or even offer new spaces to other players that grant you extra goodies when taken. Agricola has obviously had a lot of thought put into it, offering almost every way for players to interact without interacting.
The object of Agricola is to improve your 17th-century family farm over the course of 14 rounds. The players (from one to five) each have their own boards, with 15 spaces arranged in a 3x5 grid. Two of the spaces start out filled with the two rooms of a wooden house; throughout the game, the spaces get taken up by more rooms, fields for planting grain and vegetables, and pastures and stables for holding animals. So as the game continues it's like the player is maintaining this little personal simulation game.
At the end of the game, each player's farm is scored in a number of categories. Players get from one to four points for having at least one grain, vegetable, sheep, boar, cow, field or pasture. (Actually, you need at least two fields for points.) Each of these categories that doesn't score at least one point instead causes them to lose a point. Every space of their board that isn't used for some purpose also loses them a point. This means that Agricola wants players to diversify, trying to cover as many bases as they can. Yet, in the usual style, players also gain play advantages for focusing in one area. Often this means a player will pick a single area or two to focus on in the early rounds, and try to use the breathing room that offers to give them the space turns later to fill in the other areas.
While it has a lot of distinctive mechanics, I think the most distinctive is how Agricola forces players to satisfy an onerous food requirement. Players start out with two workers, and at each Harvest time, they must feed each worker they have two food tokens. (Three in a solitaire game.) Many of the resources players build can be converted to food in different ways, although other buildings must be constructed to get substantial benefit out of most of them. There are six Harvests, that happen more and more frequently throughout the game, at the ends of rounds four, seven, nine, 11, 13 and 14. Starting with round seven, harvests happen every other round, and the last two rounds have consecutive harvests. Not feeding all of your workers forces a player to take a Begging Card for each food they're short. A Begging Card is worth -3 points at the end of the game! Even taking one Begging Card is a gigantic penalty in a game as low-scoring as Agricola, and if you neglect the food requirement you'll quickly doom yourself.
Unless you gain more workers through the game, with two workers and 14 rounds, that means, you'll only have 28 moves to make during the game in total. Over that time, you'll need to scratch up 6 Harvests x 2 workers x 2 food, or 24 food in all. But to do well in Agricola, you'll need more turns than that. The main method of doing this is by growing your family, by taking Family Growth spaces. Usually, to do this, you'll first have to expand your house to hold them, which is the most costly kind of resource sink in the game. To grow your family, you have to gain five wood (or some other, even rarer resource), plus two reed, and then take the Build Rooms space, and only then can you take Family Growth for another worker, which can be put into service starting the next round. That four of your 28 moves right there. Late in the game a Family Growth (Even Without Space In Your Home) space opens up, but by that time there's at most only two turns in which to take advantage of it.
Even though having more workers means having to scrounge up even more food, in nearly every case, if they can keep that food requirement met, the player with the most family members is usually the one who ends up winning. Those extra actions are very useful, and with careful play someone can usually use them to more than make up for the increased food usage.
A lot depends on which Occupations and Minor Improvement cards a player starts the game with. Players begin with a hand of seven of each, and they can be put into play during the game by taking an Occupation space (which usually consumes a whole turn, and is thus quite costly) or a Minor Improvement space (frequently an option on some other space, especially Starting Player). Most of these cards give the player an advantage, sometimes weak and sometimes strong, that exempts them in some way from the competitive scrabble for a specific resource. This affects the game in subtle and profound ways: having a card that gives you extra sheep means you likely won't be chasing the sheep piling up on the Sheep space. That saves you the turn from having to collect it, and means that it'll be more prone to having many sheep on it when someone else takes it. It also makes the Cooking improvements more powerful for you, since you can use them to turn sheep into food, which then means the direct food collection spaces will have less competition. In a full game of Agricola, everyone has fourteen of these cards, and you won't be sure who has what, and everyone's hidden cards and strategies mix together to make a fascinating maze of unknown priorities to navigate.
The game leaves it up to the group how the Occupations and Minor Improvement cards are distributed. Three options are suggested by the rules. They can just deal seven of each to everyone, which makes for a swingy game where some players will probably have bad choices; they can deal out 10 of each type to everyone and let them discard their least favorites; or the most time-consuming options is to let them draft the cards, dealing seven of the two types to each player, letting them pick one of each kind to keep and passing the rest to their left, and in this manner build up their starting hand out of the discards of the others. Some cards are really powerful: the Taster, for example, lets that player overcome the turn order and go before the starting player, at the cost of giving that player a bit of food. A few cards like that one have the power to change the entire game, but most of them are fairly modest advantages, or are balanced by the resource costs needed to activate them.
I wish at this point I could offer some easy and inexpensive way to play Agricola on your computer, but for whatever reason it's really hard to find a good way to play it that way. There's an okay iOS version but it's locked to Apple platforms. There used to be an Android version, but it was buggy and the rights holders decided to take it off the platform instead of fixing them. The Steam version has been similarly delisted. The physical version is published by Asmodee, who has made this decision to kill those versions instead of fixing them. It's pretty awful of them. (If you've seen the game "Agricola: All Creatures Great and Small" in the game stores, it is not the "real" Agricola, but a spinoff game that plays differently.)
There is still a free public way to play it, on the site playagricola.com. At the moment, it's the best that I can offer for playing it quickly. I think, as with many board games, the best way to play is in person with a physical set, but it might be a good way to dampen your shoes. I notice that it's also available on Board Game Arena as a Premium game.