With the sentence construction: Subject, to be, adjective/adverb, as in "I am bored" or "we are tired" or "I am well", Does the predicate function as an adverb or an adjective? I think I have seen it as both, which is why I'm asking this question.

when you say "i am tired" you say "mi estas laca" but when you say "i am well" you say "mi estas bone"

Why do you have one ending with -e and the other with -a, and how do you know weather to end with -e or -a in this specific sentence construction?

Dankon

"I am ..." always will be translated "mi estas ...a", 'mi estas ...e' is impossibla.
I am well => mi estas bonstata = mi estas en bona stato = al mi estas bone = mi sentas min bone
Similarly:
I am cold = al mi estas malvarme.
But:
Ice cream is cold = glaciaĵo estas malvarma.
"It is ..." if 'it' means nothing, will be translated "estas ...e", "estas ...a" is impossible without a subject.
It is well => estas bone.

Nala,

To elaborate, what sergejm said, germanic languages (e.g. English) have a fixed word order, which is subject-verb-object (SVO; "I love you") except for questions, for which it is VSO ("Do you love me?"). English complicates things by lacking any marking for object (except for person pronouns), which is done either by the accusative case (e.g. in E-o) or a special object marking (e.g. in Japanese) in many other languages.

Before reading further you may want to take a look at a Langfocus video on Youtube.

The fixed word order and the lack of grammatical cases cause together a need for formal subject in a sentence.

• It rains : What "it"?
  
• There is poverty in the world : Where "there"?

These "it" and "there" don't mean anything, i.e. they are formal subjects, that are only needed to have the SVO order in the sentence. E-o doesn't have formal subjects – and allows other word orders (I have written more extensively about this in a previous thread here in Lernu).

The verb be (en)/esti (eo) can be used as a copula, as a linking verb, i.e. it describes the subject ("What are you? I am tired"). The descriptive word is grammatically called predicative, and predicatives are in E-o always nouns or adjectives ("Mi estas studento/Mi estas laca") in nominative.

What throw you off, is that formal subject. In the English sentence "It is well", "it" doesn't denote anything. Consequently "well" isn't a predicative, a description of the subject, but an adverbial, which describes the predicate, main verb. Adverbials in E-o are either adverbs (English "well" is also an adverb; "Estas bone") or prepositional expressions ("James respondis en la angla").

Finally as a remainder most verbs are non-copulative, and if they take a direct object (i.e. are transitive), that object is in accusative in E-o ("Mi farbis domon").

Nala_Cat15:when you say "i am tired" you say "mi estas laca" but when you say "i am well" you say "mi estas bone"

The word "well" in the sentence "I am well", is not an adverb, but an adjective. https://en.oxforddictionaries.com/definition/well and also Esperanto uses an adjective here: Mi estas sana.