Three or four players (up to 6 with the extension) play against each other to create the most prestigious settlement on the island of Catan, rolling dice to collect resources to build roads, towns, cities and other advancements. (90-120 minute playtime)
The island of Catan is made of hexagonal tiles which produce specific resources needed during the game. Three or four players (up to 6 with the extension) build settlements and roads on the corners of these tiles to gather the resources. Eventually, they each build a little empire, upgrading settlements to cities, buying development cards, and playing soldiers to fend off the marauding robber. The first one to reach 10 victory points wins. Playtime is usually 90-120 minutes.
Getting resources is dependent on the placement of one's settlements and the roll of the dice, so it is often necessary to trade with other players (or the "old world" at predetermined ratios) in order to obtain the desired advancements.
There are four expansions for the game:
seafarers-of-catan adds ships and the possibility more complex board layouts including multiple islands.
catan-cities-and-knights expands trade, adds secondary resources, reworks the development deck into 3 different progress areas, introduces barbarians that launch assaults on all of Catan and the knights used to defend against them.
catan-traders-barbarians expands the barbarians mechanic and further expands trade.
catan-explorers-pirates adds the concept of additional islands.
Originally released in Germany in 1995 as Settlers of Catan and rebranded as Catan in 2015, the game is immensely popular and is many people's first introduction to "designer" board games.
For answering questions the official rules are found here