I've been playing Civilization since launching it from DOS on Civ 1 in 1993 or so. My friend's dad gave everyone their own floppy disk to save games on and we could play on either of their computers (the new one ran Windows 3.1 and had a CD drive!). When Civ II came out in 1996, my family finally had our own awesome computer: Pentium 100MHz upgraded to 32MB of RAM and a 1GB HDD. My Brother and I eagerly picked up a copy at Guam's computer store, Megabyte Computers, and continued the quest for world domination at our own house.
Civilization 1 will always have the greatest nostalgia for me. It also came with the mystery of ruling an empire unsure of how decisions would impact my future. I feared a REVOLUTION and thought that walls somehow helped with food (I was 8 at the time). I knew that Montezuma would have his revenge against me and respected Stalin's heavy serious music. Warlord was a challenge and so I never had to worry about levels where the AI had to cheat to win.
Civilization 2 was special because it was designed (intentionally?) to be modded. My Brother created a massive scenario with all new units where the Spaniards invaded the accurately-mapped Marianas islands from Manila and you controlled the primitive chiefs and canoes in what seemed a valiant and futile resistance (at least I never beat it).
Civilization 3 was a disappointment after Alpha Centauri. The AI was too much worse to play the game until the final expansion fixed the computer's aggressiveness. Civ 4 was amazing, making multiplayer fun, fixing combat strength, but further highlighting the problem of stacking units. Civ 5 was bold, and successful in its boldness. It fixed several major flaws of its predecessors without introducing major new ones and expanded the unique feel of each civilization. However, somewhere the game lost a little of its soul.
I never go back to Civ 4. I will never stack 100 catapults again. I never liked Civ 3. I sometimes go back to Alpha Centauri. And I don't know why I don't go back to Civ 2. But I do go back to Civ 1 from time-to-time. It's not any less fun than before. I don't find myself wishing it was more complicated because it's already perfectly balanced and sometimes simplicity is the most satisfying. That is, until I lose my Armor attacking a Phalanx. So I guess the original is my favorite.
If you want to know my favorite civilization to play as, it's the Zulus in Civ 1, Vikings in Civ 2, Morganites in Alpha Centauri, Chinese in Civ 3, Romans in Civ 4, and Random in Civ 5.