Structure of "reason that I was confused"

Question

What kind of structure does the sentence below have?

The reason that I was confused is ...

Is that I was confused an adjective clause or not?

Also we can omit that and say: the reason I was confused.

Why can that be omitted?

Answer 1

This construction is idiomatic, and specific to the noun reason. A that clause following reason tells the hearer or listener what reason explains.

The reason that I was confused is equivalent to
The reason why I was confused.

Both reflect the underlying fact

I was confused because [reason].

And the reason why you may omit the that (or why) is that the clause headed by that (or why) is a special sort of relative clause; and just as in any other relative clause, with that or who or which, the "relativizer" which heads it may be omitted if it does not stand for the subject of the clause.

The car { that / which } I am driving is a Toyota.
The reason { that / why } I was confused is that English is a horribly complicated language (just like all languages).

Answer 2

Is that I was confused an adjective clause or not?

It is a relative clause; it begins with a relative pronoun (such as that, for example) and defines/identifies the noun that precedes it (reason in this case).

Why can that be omitted [from the reason that I was confused]?

Because it is the object of the relative clause, one of the two occasions when it is proper to do so. There's more information about this on this other web page.