View Solution

To get this achievement you need to train at least one of every unit type in the base game, Gods and Kings and Brave New World units are not required for this achievement. Note that you must train each unit in a city (i.e. starting units, purchased units, gifts from city-states, upgraded units, etc. don't count). I recommend pursuing this in tandem with:
(check out my guide and just swap in whatever era you need to be in for your unique units!) Here is a list of all the units required and the tech required to build them:
Ancient Era
Archer: Archery
Chariot Archer: The Wheel (requires Horses)
Hoplite: Bronze Working (Greek unique unit, replaces Spearman)
Immortal: Bronze Working (Persian unique unit, replaces Spearman)
Jaguar: No tech requirement (Aztec unique unit, replaces Warrior)
Scout: No tech requirement
Settler: No tech requirement
Spearman: Bronze Working
Trireme: Sailing
War Chariot: The Wheel (requires Horses, Egyptian unique unit, replaces Chariot Archer)
War Elephant: The Wheel (Indian unique unit, replaces Chariot Archer)
Warrior: No tech requirement
Worker: No tech requirement

Classical Era
Ballista: Mathematics (Roman unique unit, replaces Catapult)
Catapult: Mathematics
Companion Cavalry: Horseback Riding (requires Horses, Greek unique unit, replaces Horseman)
Horseman: Horseback Riding (requires Horses)
Legion: Iron Working (requires Iron, Roman unique unit, replaces Swordsman)
Mohawk Warrior: Iron Working (Iroquois unique unit, replaces Swordsman)
Swordsman: Iron Working (requires Iron)

Medieval Era
Camel Archer: Chivalry (requires Horses, Arabian unique unit, replaces Knight)
Chu-Ko-Nu: Machinery (Chinese unique unit, replaces Crossbowman)
Crossbowman: Machinery
Knight: Chivalry (requires Horses)
Landsknecht: Civil Service (German unique unit, replaces Pikeman) Note: version in Brave New World does not count towards the achievement as it cannot be trained, only bought. You will need to disable BNW. See the end of this solution for how to do this.
Longbowman: Machinery (English unique unit, replaces Crossbowman)
Longswordsman: Steel (requires Iron)
Mandekalu Cavalry: Chivalry (requires Horses, Songhai unique unit, replaces Knight)
Pikeman: Civil Service
Naresuan's Elephant: Chivalry (Siamese unique unit, replaces Knight)
Samurai: Steel (requires Iron, Japanese unique unit, replaces Longswordsman)
Trebuchet: Physics

Renaissance Era
Cannon: Chemistry
Caravel: Astronomy
Frigate: Navigation (requires Iron)
Janissary: Gunpowder (Turkish unique unit, replaces Musketman)
Lancer: Metallurgy (requires Horses)
Minuteman: Gunpowder (American unique unit, replaces Musketman)
Ship of the Line: Navigation (requires Iron, English unique unit, replaces Frigate)
Sipahi: Metallurgy (requires Horses, Turkish unique unit, replaces Lancer)


Industrial Era
Cavalry: Military Science (requires Horses)
Cossack: Military Science (requires Horses, Russian unique unit, replaces Cavalry)
Rifleman: Rifling
Artillery: Dynamite
Ironclad: Steam Power (requires Coal)

Modern Era
Anti-aircraft Gun: Ballistics
Battleship: Electronics (requires Oil)
Carrier: Electronics
Destroyer: Combustion
Foreign Legion: Replaceable Parts (French unique unit, replaces Infantry/Great War Infantry)Note: version in Brave New World does not count towards the achievement as it cannot be trained, only obtained through the Freedom tenet. You will need to disable BNW. See the end of this solution for how to do this.
Infantry: Plastics
Submarine: Refrigeration

Atomic Era
Anti-tank Gun: Combined Arms
Fighter: Radar (requires Oil)
Panzer: Combined Arms (Requires Oil, German unique unit, replaces Tank)
Tank: Combined Arms (Requires Oil)
Zero: Radar (requires Oil, Japanese unique unit, replaces Fighter)
Atomic Bomb: Nuclear Fission (requires Manhattan Project national wonder and Uranium)
B17: Radar (Requires Oil, American unique unit, replaces Bomber)
Bomber: Radar (Requires Oil)
Helicopter Gunship: Computers (requires Aluminum)
Mobile SAM: Rocketry
Paratrooper: Radar
Rocket Artillery: Rocketry (requires Aluminum)

Information Era
Guided Missile: Advanced Ballistics
Jet Fighter: Lasers (requires Aluminum)
Mechanized Infantry: Mobile Tactics
Missile Cruiser: Robotics
Modern Armor: Lasers (requires Aluminum)
Nuclear Missile: Advanced Ballistics (requires Manhattan Project national wonder and 2 Uranium)
Nuclear Submarine: Telecommunications
SS Booster: Advanced Ballistics (requires Apollo Program national wonder and Aluminum)
SS Cockpit: Satellites (requires Apollo Program national wonder and Aluminum)
Stealth Bomber: Stealth (requires Aluminum)
Giant Death Robot: Nuclear Fusion (requires Aluminum)
SS Engine: Particle Physics (requires Apollo Program national wonder and Aluminum)
SS Stasis Chamber: Nanotechnology (requires Apollo Program national wonder and Aluminum)

So there you go: 79 units in all! Note that 2 of these units are no longer trainable in Brave New World: the German Landsnecht and French Foreign Legion. To obtain these you will have to start a game with BNW disabled. To do this click on DLC in the Main Menu and then uncheck Brave New World. You will now be able to train these units when playing as their respective civs.

Also, if you're not sure what units you still need for the achievement (or any others) you can check by enabling the achievements_debug log. To do this go to My Documents/My Games/Sid Meier's Civilization 5 and open config.ini with your preferred text editor. Find the line that says LoggingEnabled = 0 and change the 0 to 1. Now the next time you start the game a file called achievements_debug.txt will be created in the logs folder. If you scroll down some all the units you need, and how many you have already trained, are listed between the lines Steam Stat: ESTEAMSTAT_ARCHER = # and ESTEAMSTAT_SSSTASISCHAMBER = #