Carcassonne is a truly wonderful game. Players take it in turns to place tiles and claim features such as cities, roads, cloisters and farms by choosing whether to deploy a follower from their supply each time they place a tile. Points are awarded when claimed features are completed depending on the size and type of feature, and any followers that occupied it are returned to their owners so that they can be reused elsewhere. The game continues as such until all tiles have been placed and a unique landscape has developed as a result.
It sounds simple and it is, but it can become very tactical and competitive once you've played a few games. Farms are slightly more complicated than any of the other features but it doesn't take long to get used to them and if your games of Carcassonne are anything like mine they'll usually have descended into an all-out war for control of the most valuable farm by the end!
With a bit of knowledge of the number and configuration of available tiles it's possible to make your opponents features impossible to complete if the required tile either doesn't exist or has already been placed elsewhere. This means their follower becomes trapped and this reduces their scoring potential for the rest of the game (spouses don't like this in my experience!) It's also possible to place tiles such that features owned by different players are joined together and become a single road/ city. If more than one player has a follower in such a feature then the player with the majority of followers takes the full amount of points for that feature when the feature is closed regardless of who originally claimed it. So stealing potentially lucrative features is another great tactic and something else to consider.
To describe Carcassonne merely as a good "gateway" game is to deny it the credit it so rightly deserves. I'm partly guilty of this as I saw it this way at first and "moved on" to some of the other games that I had become aware of since realising that there's more to board games than Monopoly and Risk. I enjoyed most of them well enough, but continually returned to Carcassonne as I found it to be far more enjoyable than anything else I'd played. Clearly the fact that tiles are drawn at random means that there is an element of luck in this game. The best players however are the ones who can lay their tiles and deploy their followers in such a way that they're able to use whatever tile they draw to score points or other improve their position.
Furthermore, it's worth remembering that the basic game can be significantly expanded using any of the numerous (about 30) expansions available for it. As other reviewers have mentioned, Inns & Cathedrals and Traders & Builders are generally considered to be the best although games can start getting somewhat more complicated when played with several expansions at once. The River (included in this box) is also a great starting point for those new to Carcassonne as it enlarges the starting area from one tile to 12 and thus provides players with a wider choice when it comes to placing their tiles.