BUT WAIT! That can't be correct, because once we drop the "it" that adjective doesn't have a noun (or pronoun) to hang off of.

Spot on.

The usage of "esti terure", "naĝi estas sane" and "estas bone, ke vi venis" is obviously a calque from Russian (and German) syntax.
Both languages (known to Zamenhof) don't have a proper adverbial form like English -ly[1]. When you need to turn an adjective into an adverb you just take its neutral form and run for it. Now this neutral form maps in Esperanto to -e.

This is how I think about it:
An Esperanto adjective comes in five flavours:
-a: marking singular and nominative,
-aj: marking plural and nominative,
-an: marking singular and accusative,
-ajn: marking plural and accusative,
-e: marking neither number nor case.

Now when it is used predicatively (e.g. with esti), the adjectives tries really hard to agree with its subject in both number and case.
La hundo estas juna.
La hundoj estas junaj.

However when there is no subject, it cannot agree with it and falls back to its simplest form, marking neither number nor case.

Estis terure.

When the subject is an infinitive or a subclause, which both don't have neither number nor case, the same thing happens. The adjective feels completely bamboozled and falls back into its most basic pattern -e.

Naĝi estas sane.
Ke vi amas vin, estas bele.

- - - -
[1] which in English originally wasn't adverbial neither, but just a morpheme for deriving adjectives from other parts of speech (as in friend -> friendly). Compare German -lich.

To add a little to the good explanation of RiotNrrd…

In Esperanto an adjective describes a noun, a number or another adjective, although the last two cases are quite rare:

• terura vetero : terrible weather
  
• terura tri : terrible three

To describe anything else, whether it is a verb, a sentence whatever, you use adverb:

• terure dolori : to have a terrible pain
  
• Estis terure, ke… : It was terrible that…

Yes, the linguistic term for the adverb with the verb esti is predicative as Nornen said.

A problem with many dictionaries like bab.la is that they tend to give translations for single words or word pairs without giving proper explanations and examples. There is a recent thread about resources which to use. Treat other online resources with suspicion.

Also the e-form is the multi-tool of Esperanto. It isn't just an ad-verb, it is a freaking ad-anything. While adjectives can only modify nouns and pronouns, those ad-anythings can modify just about everything.

Can it modify a verb? Surely, that's why it's called an adverb:
Mi kuras rapide.

But can it modify an adjective? Yes, it can:
Mi estas nekredeble rapida.

How about another adverb?
Mi kuras nekredeble rapide.

Can it modify a whole sentence?
Hieraŭ la vetero estis bela.

How about a noun? Surely not you say? Hold my beer:
Ĉu vi vidas la arbon tie?

Can it modify nothing at all? It surely can:
Certe.

That is a totally awesome illustration of the use of adverbs.

English tends to favor adjectives. Esperanto is much more adverb heavy.

For structural reasons adverbs show up in all sorts of places in Esperanto that they don't in English. This is another reason why it can be important to break the association of adverbs with -ly words in your head. You will hit Esperanto adverbs that just don't fit that pattern, places where they are being used in non-English ways, and it can be completely confusing unless you get what adverbs in Esperanto are doing functionally. So pay attention to any rules, tips, and hints about adverbs whenever you run across them. In Esperanto adverbs are more heavily relied upon than in English and fulfill a lot of different roles, and mastering them will take you a very long ways.

RiotNrrd:In the case of the general sentence: "It is terrible."

We've had this conversation before in relation to the weather, but here it is again. If the sentence is meant as a comment on the general state of things, the word "it" in that sentence doesn't refer to anything. English requires something in that position, so we stuff an "it" in there. But "it" isn't actually pointing at anything. It's grammatical filler.

In Esperanto, we don't do that. Our pronouns have to refer to things. So we drop the meaningless "it", leaving us with the impulse to just use the verb and the adjective. "Estas terura". BUT WAIT! That can't be correct, because once we drop the "it" that adjective doesn't have a noun (or pronoun) to hang off of. Adjectives are descriptive words for nouns, and therefore MUST have nouns that they describe. "Estas terura" lacks a noun. So, no nouns, no adjectives. But it does have a verb (estas), which is literally the only thing we have left to work with. What modifies (or describes) verbs? Adverbs.

"Estas" by itself refers to an ongoing state (or technically act) of being (i.e., "how things are going"). So in our sentence, what is that state of being? What word describes that state? How are things going? Well... terrible, whose root is "terur". "Terrible" + the adverbial marker (required to note that it is describing the verb, in this case "estas") = "terure".

So the correct expression is "Estas terure". It has a verb, and an adverb to describe that verb. All the parts work together properly.

The alternative is to make the whole thing a single word: "Teruras". This combines the verb and its description into something which for all practical purposes means the same thing as "Estas terure". This is also generally, I believe, the dominant method of this kind of expression.

Do NOT literally translate Esperanto adverbs by throwing an -ly on the English version. That only sorta works, and sometimes just generates nonsense. "Estas terure" does not mean "Is terribly", it means "things are going terribly" or, more shortened, "It is terrible". English adverbs and Esperanto adverbs only work similarly. They are not the same, and thinking of them as the same will only confuse you. Pay attention to the function of the words, not how they map to English. This becomes doubly important when you are dealing with words that have no English equivalents, and therefore no -ly forms to glance at for inspiration.

Adverbs describe verbs, and in some situations can stand alone. Adjectives do not work this way at all, and must have a noun to attach to. If you have a sentence with just a verb and an adjective, the sentence is improperly formed. You cannot describe a verb with an adjective.

Useful post. Thanks.

A few things:

First, you mentioned that this was ground we already went over to some extent. I don't remember the weather conversation, but I will say this: I expect that you guys are going to have to tell me things multiple times before it all sinks in. I think that as we go over stuff, some fraction of it hits home and gets remembered while the rest doesn't quite sink in. But the hope here is that through repeated exposure and hitting topics from different angles, all the fractions are going to add up to something resembling a full understanding.

Second, here is the fraction from this lesson that I think will get remembered: Don't adjectives without an explicit noun. That's going to be a helpful guideline to remember going forward, and I do feel like my understanding is more clear now. Things are still foggy, but you have dispelled the fog somewhat and allowed me to see more than I could before.

Third, the use of verbs overall is weird to me in Esperanto. "Teruras" to me does not seem like a complete sentence. As I look at it I am asking, "Where's the rest of it?" But I can kind of get it. Though I'll mention that when I look up "Teruri" in the dictionary (both here and at bab.la) I get "to frighten, terrify." So can Teruras really mean "It is terrible" in the sense that "It is really unpleasant" or "It is really low-quality"? It seems like it would mean "It is terrifying" or "It is horrifying."

Metsis:To add a little to the good explanation of RiotNrrd…

In Esperanto an adjective describes a noun, a number or another adjective, although the last two cases are quite rare:

• terura vetero : terrible weather
  
• terura tri : terrible three

To describe anything else, whether it is a verb, a sentence whatever, you use adverb:

• terure dolori : to have a terrible pain
  
• Estis terure, ke… : It was terrible that…

Yes, the linguistic term for the adverb with the verb esti is predicative as Nornen said.

A problem with many dictionaries like bab.la is that they tend to give translations for single words or word pairs without giving proper explanations and examples. There is a recent thread about resources which to use. Treat other online resources with suspicion.

Okay, noun, number or another adjective. I'll try to remember that.

Can you provide an example where an adjective describes another adjective? I can't think of anything like that.

nornen:Also the e-form is the multi-tool of Esperanto. It isn't just an ad-verb, it is a freaking ad-anything. While adjectives can only modify nouns and pronouns, those ad-anythings can modify just about everything.

Can it modify a verb? Surely, that's why it's called an adverb:
Mi kuras rapide.

But can it modify an adjective? Yes, it can:
Mi estas nekredeble rapida.

How about another adverb?
Mi kuras nekredeble rapide.

Can it modify a whole sentence?
Hieraŭ la vetero estis bela.

How about a noun? Surely not you say? Hold my beer:
Ĉu vi vidas la arbon tie?

Can it modify nothing at all? It surely can:
Certe.

So if that's the case, it sounds like "adverb" is not even the right term for -e words in Esperanto.

And what I can't understand them is this: Are they a logical part of the Esperanto language puzzle, governed by rational rules that make sense, or are they just a wildcard that can do whatever they want, like a Joker in a card deck?

Though I'll mention that when I look up "Teruri" in the dictionary (both here and at bab.la) I get "to frighten, terrify."

So you cannot say "teruras" instead of "estas terure"

And what I can't understand them is this: Are they a logical part of the Esperanto language puzzle, governed by rational rules that make sense, or are they just a wildcard that can do whatever they want, like a Joker in a card deck?

Simple rule, PM-K., if you want to qualify a noun or pronoun then use the a-form. For anything else use the e-form.

Exception: when by context or convention the missing (subkomprenita) noun/pronoun is obvious , then you can use the a-form although the noun/pronoun which is being qualified is absent. So this isn't really an exception

examples:
Ĉu vi parolas la anglan? Do you speak English? (what else could the missing noun be other than 'lingvon')
Cu vi volas la ruĝan? Do you want the red one? (said in situations where it is obvious what items are under selection)

Might be different opinions about this one

Mi trovis necesa doni kelkajn klarigojn ... (missing tion)