An Aesop is a given work's ethics lesson; it's the whole point of the work. Usually, it's shown to the audience in such a way as to not interfere with watching the show or reading the book, and then it might be stated or made more clear by the end of the work. Ethics philosophers might be referenced and famous thought experiments might be shown, but it's usually subtly done so as not to interfere with the story. We generally just call these morals, and if they're ever stated outright, they're likely in the forms of aphorisms. Here on TV Tropes, we call these aphorisms "Aesops."
An "anvil," meanwhile, is a heavy-handed ethics lesson. In fact, it's so heavy-handed that you could say it's like getting hit with an anvil. Hence the name.
Anvils are tools, first and foremost. They could be "done poorly" and they could be "done well," but at the end of the day, an anvil is simply what it is: a clumsy ethics lesson that insists upon itself.
That said, just because it's clumsily done does not mean it's an anvil. It must be (1) some kind of ethics lesson and (2) heavy-handed. So, the creator must actually give the audience something to be learned, like "Slavery is wrong" in Uncle Tom's Cabin. It must also be heavy-handed, like stopping the story in order to explain the evils of slavery in Uncle Tom's Cabin.
While the lesson doesn't need to be "successful," it does still need to be a coherent lesson. Thus, if the work contradicts its own Aesop, then it isn't an anvil but rather a Broken Aesop. Moreover, the lesson must be heavy-handed, so if the Aesop isn't intentional, it isn't quite an anvil but an Accidental Aesop. In addition, a Lost Aesop is the total opposite of an anvil, as it's done so subtly that it might not even exist.
Other types of Aesops are prone to this, however, such as Captain Obvious Aesop, Scare 'em Straight, and Hard Truth Aesop. Basically, if the writer is climbing uphill, then they may be inclined to make an anvil rather than an Aesop.
Sister trope to An Aesop. Contrast Lost Aesop. Specific aphorisms can be found on Stock Aesops. Supertrope to Anvilicious and Some Anvils Need to Be Dropped. See also Author Tract and Propaganda Piece.