In no particular order:
England
UA - Sun Never Sets: This is a great quality of life ability with a surprising number of uses. Not only does it help you get around, but it makes for quicker invasion, quicker reinforcement of your cities, and quicker retreat for wounded units. The extra spy helps you rocket through the top of the tech tree after you're done scraping the bottom for Longbowmen and the Leaning Tower.
UU1 - The Longbowman: Possibly my favorite UU, the Longbowman is just a whole lot of fun to war with. I try to keep peaceful until the middle ages, when Longbowmen scour the world with arrows before the enemy can even reach them. I sometimes wait to upgrade them, but the English Gatling Gun is like a 3rd UU, even stronger than the UU it upgraded from.
UU2 - The Ship of the Line: Longbowmen lose some of their sparkle once you have to leave your continent for conquest, and that's where the Ship of the Line comes in. Of England's 3 UU's, the Ship of the Line is my least favorite, and I love the Ship of the Line.
England is, bar-none, my favorite military Civ. As soon as the middle ages come, England keeps conquering until none are left to conquer. The game always stays interesting as England, because you get a new toy every age - first longbowmen, then two spies, then Ships of the Line, then super gatlings. Plus, I love Elizabeth I. You go, Gloriana.
Byzantium
UA - Patriarchate of Constantinople: Possibly my favorite UA, though the other civs in this list (except for England) give it a run for its money. Customizing a religion is one of my favorite parts of the game as I really like to imagine what my empire would be like, and no one does it like the Byzantines. You can do a lot of fun, creative, and powerful things with their UA - tithe/church property, itinerant preachers/religious texts, super-temples, the list goes ever-onward. I play the Byzantines a lot because I'm always thinking of how new combinations might work. For example, once I can download the new patch (curse you Macbook!), I'm excited to try the Byzantines with Interfaith Dialogue, Pilgrimage, and Holy Order.
UU1 - Dromon: Ah, the joy of having something your neighbors don't. The Dromon is nice for exploiting the honor opener, and it's great for early invasions, especially when a city is coastal and covered by rough terrain. In fact, its Byzantine brother-in-arms is optimized for the same situation.
UU2 - Cataphract: I just feel cool using the Cataphract, and they're extremely effective in early wars. Cataphracts thrive in cities that are normally tough nuts to crack. Their extra movement point lets them get into position without any trouble, and their defensive bonuses let them stay there until your dromons and your siege engines take down the city's health, when they can sweep in for the kill.
The two things that give me the biggest kick in this game are getting creative with a Civ's abilities and doing it with style. That's one of the reasons I almost never play as Egypt - their UA, UB, and UU are all tailor-made to my playstyle, but I just cannot stand the color yellow. Say what you will about Theodora, but she's the most stylin' Queen in the game. Dragon-ships, tank-knights, and a super-religion? Sold.
Korea
UA - Scholars of the Jade Hall: An awesome ability that ensures you'll have a massive tech lead if you follow the jade-brick road - go for Writing ASAP, and try to get a Library built before you construct the Great Library for 3 tech boosts from one-tech. Nebby's ingenuity has nothing on the scholars of Sejong. Plus, I love specialists and I love great people, and they're tastier than ever when they're Korean. It's also fun to do a 3- or 4-city culture game as Korea, focusing on artist specialists. It's a good way to keep up with tech as you go for Utopia -- Oppa Gangnam Style.
UU1 - Hwacha: Hwacha doin'? Oh, you know, Korean stuff I guess. I like the Hwacha well enough, but frankly I'd rather have Trebuchets. They are great defensive units, though.
UU2 - Turtle Ship: The Turtle Ship is just super cool, especially on a map with lots of water, like Small Continents or Archipelago. No coastal city within the turtle ship's reach is safe. Though I must say, the only thing better than having turtle ships is stealing them with English privateers. The English are just so good at stealing things. But I digress.
Korea is by far the strongest science Civ, but that's not why I like them. Once again, it all comes down to style. I love the flexibility of their ability. Not only do you have extremely powerful specialists (the Rationalism tree and the Statue of Liberty are musts), but you have constant opportunities to exploit it in creative ways, such as using the GG from Honor as a source of early science or with artist-scientists. Also, Sejong is definitely among my top 5 favorite kings of all time. If they ever make a Brazil civ with Dom Pedro II I will die of happiness. If they make a Brazil civ without him I will die.
The Aztec
UU - Sacrificial Captives: Grab autocracy and win a cultural victory by SACRIFICING THEM ALL TO THE GODS. No one's more metal than Monty. Or you could use the ability for peace - you need a sizable military to ensure peace with your neighbors, but a sizable military is costly. To get returns on your investment, you need to go to war. It's the eternal problem. But you don't need to go to war when you're Monty - even when all your fighting is barbarians, your military can be more than an expensive measure of protection. It can be a direct, steady, and effective exchange of gold for culture.
UU - Jaguar: The jaguar is such a great unit. I almost always build a scout first, but jaguars are exceptions to the rule. You're encouraged to build them early while you can, and that means you have a strong enough military to harass barbarians, bully city-states, and keep the homeland safe even in the early game. Hang onto them and you have an elite, mobile, self-healing army on your hands that lasts throughout the ages.
UB - Floating Gardens: Definitely one of the best UB's, the only shame is that you're screwed out of it if you don't settle on a river. The more food bonuses you can stack with it the better. Hanging Gardens are a priority, and thematic anyway.
The Aztecs are one of the best all-around Civs in the game in my opinion, and their abilities are very cohesive and very powerful. You end up with a monstrous capital and a monstrous army to beat.
My 5th choice is a tough one between France, Polynesia, Sweden, Carthage, and America. I've already ran my mouth off so I guess I'll leave the last one at that.