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human
ELI5: What's the difference between lager, ale, stout etc?

When you go to a bar there are always lots of different drafts and bottles of beer. What makes them all different?
gpt
Starting with the "beer" category, you have two kinds: Ales and Lagers, which are the two types of fermenting yeast. Ales are fermented warm,  the yeast sits on top, and it happens in a matter of days; Lagers are a relatively new (past couple hundred years) type of yeast that ferments cold, on the bottom of the fermenter, and takes weeks. 

Lagers are typically (though not always!) a lighter style, like Budweiser, Carlsberg (the scientific name for lager yeast is actually "Saccharomyces carlsbergensis"), and so on while Ales have a more complex range of outcomes.

Now getting to the next categorizations, you have a few within each Lager and Ale. Lager has fewer - generally you're looking at a pale lager like a Budweiser, but there are also black lagers and a bunch of pale types like Helles. Within ales, you have stouts, porters, amber ales, golden ales, list goes on. 

To help explain these categorizations, it's helpful to understand what makes one beer taste different from another. There are generally 4 or 5 ingredients to beer: Water, Malt (Barley, wheat, etc.), yeast, hops, and in some other beers a cheap source of fermentable sugar like Rice or Corn (e.g., in Budweiser it's rice). 

Malt gives the beer its color. Caramel and Crystal malt, the pale kinds, aren't roasted very long and end up with golden and amber colored beer. More roasting means darker, which end up in ruby, brown, black beers. Malt gives beer flavors like chocolate, coffee, sweetness. Hops are what gives beer its bitterness and much of its aroma, and flavors like citrus. It's the combination of malts and hops that end up dictating style.

Stouts, as you can probably guess, are ales, dark malts, and the hops vary depending on what sweet/bitter balance the brewer is going for. Pale lagers are lagers, light malts, light hops. Amber ale is medium roasted, often lots of hops. Hefeweizen is wheat malt, an ale, a bit of hopping. So, you can see where this is going. Some other brews will have other ingredients like fruit, and some brews like Lambics will have weird wild yeast, but this should help you distinguish when you're at a pub.

tl;dr: The "ale / lager" distinction is settled by the type of yeast, the style within ale / lager is dictated by malt type/treatment and the hop character.