This is a pretty vast question, as there must be at least a dozen expansions for Carcassonne now, ranging in style from the pretty logical to the completely whimsical (hello Catapult). I'm afraid that I don't own ALL the expansions to Carcassone - if anyone here does, I salute them - so I can't give a truly definitive answer, so I'll give a brief description of what I think the various expansions in my old Big Box add. And that should definitely be a good starting point.
The River is a small and pretty nonessential expansion. Essentially it provides a new introductory phase to the game: instead of plunging the players right into the thick of random tile draw, they get a dozen fairly gentle and non-confrontational tiles to ease them into things. This is probably good if you're playing with beginners; as a Carcassonne veteran I can't say I would really miss it. I guess the main difference it makes is forcibly opening up the board the beginning, instead of starting players off all in the same area.... creating a play environment that's less instantly competitive.
Inns & Cathedrals introduces Inns and Cathedrals, of course, but its main contribution is really the Big Follower, which makes for a much MORE cutthroat and competitive game. Without the Big Follower it's really quite hard to muscle in on someone's city and steal all their points from under their nose. With him it's a constant threat. The inns and cathedrals add tiles to the game that basically double the value of roads or cities if you complete them... or give you nothing if you don't complete. This kind of thing obviously appeals to gambling types... and also people who love to sabotage the overly ambitious projects of others! I personally think the Inns tend to have a fairly minor impact on the game, but the Cathedrals are crucial - if one or more players start constructing a vast urban megalopolis that stands to gain them dozens of points, the chance to build a Cathedral and reduce all their plans to nothing - or possibly give them an insuperable lead, if you've underestimated their skills! - adds so much extra tension and strategy to the game.
Traders & Builders is another seemingly small expansion that in fact adds a ton of interest and options to the game. Traders give players a strong incentive to complete each other's cities - something that would be ridiculous in the basic game. The pig, like the inns, I can take or leave - it enables a few more points to be earned on top of a farmer dominated strategy, as long as you can maintain control of your vast pasture, that is! Builders though are massive - with a builder in play you can get two turns for the price of one every time you expand a city or a road with a builder on it. This is such a major effect that everyone has to try their damnedest to keep their builder in action - while trying to shut down other people's builders at the earliest opportunity! As you can probably tell by now, I really like expansions that increase the potential for interaction and aggressive competition in Carcassonne. Some people (my wife for instance!) hate games that feel too competitive - if you'd rather just work away on your Carcassonne game in one corner of the board, while the other players leave you to it, then you should take any recommendations by me with a pinch of salt...
I'm going to get a bit less verbose from here on in, because for me, the "ideal" Carcassonne setup is Inns, Cathedrals, Traders and Builders, with or without the River. However my box does also contain:
The Princess & The Dragon, which will appeal strongly to those who like a storytelling component to the game. This expansion, for me, feels less tactical and more about the "coolness" appeal of a giant red dragon rampaging over the board eating meeples. (I do really love the sheer size of the dragon meeple!)
The Tower always struck me as a pretty bad expansion, and I believe it's been retired from more recent editions of the Big Box than mine. Basically you're building high towers over the board: the higher your followers are, the further they can "reach" to kidnap and hold to ransom opposing followers. This always seemed really abstract and counterintuitive to me; it didn't seem to work especially well in the two player game that I tried it in, though it might be more fun in a many-player game. The expansion also contained a completely non-essential Tower component in which you can stack your game tiles, which I resented (presumably) being charged extra for!
Of the other expansions: I own "The Count", but have actually never played with it, as it seems to have a really complicated game of political manoeuvring and oneupmanship in it, and quite often I want to play Carcassonne because it's fairly simple and elegant! One expansion that I don't own and have played with only once, but which I was favourably impressed with, was "Abbey & Mayor". In particular I remember it introduced Abbey tiles which can be used to fill in difficult gaps... independently of whether cruel players have boxed you in with near-impossible demands on the final tile to complete your city! This is obviously a huge change and one that no one can afford to ignore; I love expansions that are that game-changing. Additionally, there was also something (Barns I think?) which enabled farms to be scored, and farmers to return to people's hands before the end of the game, which has similar ground-breaking implications on potential strategies for the game. The next Carcassonne expansion I buy will definitely be this one.
One final word how the expansions change the game: all of them contain a few or quite a lot of new tiles, which are much more complex than ones found in the basic set. In the basic set, you can "trap" people's followers pretty effectively by making it impossible for them to find the right tile to complete their city. Once you add in a few expansions, there are enough truly wacky tiles out there that, no matter what crazy limitations you place on a space, there's probably something out there that can fill it. (Or an Abbey!) The expansions therefore add an element of randomness to proceedings in that you can never be sure if a city is doomed or if you will draw into the perfect mad piece to finish it. Some might not like that element of luck but I think most quite enjoy the possibility that, against all the odds and the machinations of the other players, the gods may smile and deliver that one-tile-in-a-hundred into their lap :D