The infamous "spearman vs. tank" battles that resulted in the icon on Civfanatics - it's a burning tank with a confused Emoticon next to a rejoicing spearman.
Civ IV features Leonard Nimoy narrating a famous quote each time you research a technology. When you research satellites, he quotes Sputnik: "Beep... beep... beep... beep...."
Great People-related anachronisms can be quite hilarious. "Li Bo (Great Artist) was born in Tenochtitlan", while already rather weird, wouldn't have looked nearly as insane if Li Bo wasn't wearing a white Elvis suit.
"Hernan Cortes (Great General) was born in Tenochtitlan!"
J Edgar Hoover, Great Spy, in a ninja suit.
Muhammad (Great Prophet) is used to establish Christianity in 600BC
In Civ V, the AI has various traits between 1~10 scores. Different leaders had different "tendencies" in these scores which vary between games by 1-2 points in either direction. For example, America's likelihood to conquer city-states is an average 4 but between games can vary between 2 and 6 in value. Mahatma Ghandi's "Use Of Nukes" score in Civ 5 is a hard-fixed 12! In a 10-rank system!
Of course, the bug from the original game set Gandhi's aggressiveness at 1, but if it's lowered beyond 0 it roles over to 255!
The names of the achievements in Civ V.
If you fail to build enough wonders before the other cultures do (once another culture finishes a wonder, you lose the chance to finish yours), you earn an achievement called "Seriously?!"
A (never implemented) achievement is called "Cat Fight". You get it by being a female leader and declaring war on every other female leader on the map at once.
The “mech” is a giant bipedal mechanical robot armed with a variety of missiles, Gatling gun arms, laser-beam eyes, and a deadly kung-fu punch. Although not currently found on the modern battlefield, no doubt across the world governments and evil corporations are secretly developing these metal behemoths even as we speak. When these mechs come into service, all of the world’s other weaponry will become obsolete, and humanity’s fate will be decided by a climactic battle between armies of good and evil mechs in the streets of downtown Tokyo. (Assuming that the giant radioactive monsters don’t get us first.)
Kinda dark, when a humble worthless puny pathetic civilization destroyed in the players superior awesome technologically advanced military for the sake of resource or just for -*ehem- "fun", just want to make you laugh maniacally evil.
Even when they instigate and crash their army into your superior defenses, it can still evoke this. Having too many delegates in the World Congress and taking unfair advantage of it is also pretty funny.
A great moment while playing II was when the foreign adviser told me to seek other cultures and learn what I can from them. And then I should eliminate them. The military adviser then chimed in with an Evil Laugh.
In V, find a city state at total war with a major civ. Gift the city state a lot of highly advanced units, and they may be able to conquer the major civ's cities, keeping the ones with wonders and burning the rest to the ground.
"You have discovered scrolls of ancient wisdom." "Robotics" (Or for some meta fun "Writing")
After this easter egg occurs, you can find a unit called Jaimo who is based off of one of the staff for Civ III. He attacks by slapfighting and does disco dancing when he wins. Here is a thread about it.