Assuming you're not trolling

American Cheese is basically cheese (usually cheddar, though there's a couple other cheeses it can be made from) that's had an emulsifier (usually sodium citrate) added to it, which is an emulsifier that keeps it from separating when melted (if you've ever melted regular cheese you may have noticed that it gets kind of oily/greasy because the fat in the cheese separates out) which makes for a really nice creamy/gooey texture.

It's solid (though usually fairly soft) and comes in a block or pre-sliced.

There's sort of a few different "levels" of American cheese, with different labeling requirements depending on how much other stuff has been added to it.

At the top end you have stuff that can be labeled as "pasteurized process American cheese" which is basically what I described above. Cheese + emulsifier, and sometimes a little added fat, salt, acid, and color. It's basically cheddar that melts nicer. There's of course a range of quality here, some start with a better quality cheese than others, and/or add more or less emulsifiers/salt/fat/acid/color, but overall these are actually pretty good. If the label actually says "American cheese" on it, this is almost certainly what you're getting.

A step down you have "pasteurized process American cheese food" which again have emulsifiers and such, and are at least 51% cheese, the rest is made up of other dairy products.

Then you have things that don't even meet that definition and contain other additives, there's not really an officially regulated term for these things, and just about anything goes for them, they might be labeled as a cheese "product," "snack," "dip," "sauce," "spread" or any number of other things. These are your cheez wiz, Kraft singles, Velveeta, etc. And this is where a lot of people get the idea that American cheese is crap, some of them don't even taste much like cheese, their texture can be weirdly waxy/plastic-y, they have all kinds of weird additives and non-dairy ingredients, I once bought some bottom of the barrel cheese product singles that didn't even want to melt on my burger.

For those second two categories, you're probably not going to find the words "American cheese" anywhere on the package. You might find "American" in big friendly letters, but you'll have to look elsewhere on the package to find that it's a "cheese food/product/etc"

For the first two, you'll often find them being sliced off of a giant block at the deli counter, though some of them may be pre-packaged in the dairy aisle, for the last category you're almost never going to find them at the deli and it's all gonna be in the dairy aisle or even on an unrefrigerated shelf somewhere.

And if you're adventurous in the kitchen, you can do some pretty cool things with American cheese, those emulsifiers in them can make some magic happen with cheese sauces and such that you really couldn't do otherwise. As an example, I don't really like most seafood, but my wife loves sushi, so I sometimes make some weird non-fish sushi so we can both be satisfied. One time I made a "Philly cheese steak" roll with beef and caramelized onions, and I whipped up a sort of "Japanese cheez wiz" to go with it by making some dashi and melting American cheese into it (I think I also added some miso) and it was fucking delicious, and you wouldn't be able to do that with regular cheese without adding extra emulsifiers, the cheese and the broth just wouldn't mix right.