Learn Icelandic: Hin tignarlega kóróna er mjög falleg. - The majestic crown is very beautiful.

Both mean the same thing, but using the free-standing article Hin followed by a weak adjective and a bare noun is a formal, literary, or poetic style. In everyday spoken Icelandic, you would normally just say tignarlega kórónan.

Why doesn't kóróna have a definite article attached to the end of it here?

Because the word for "the" is already provided by Hin. When you use the formal free-standing article, the noun itself must remain bare. Saying hin tignarlega kórónan would be a grammar mistake, similar to saying "the majestic the crown" in English.

Why does the adjective tignarleg become tignarlega in this phrase?

Any adjective immediately following a definite article like Hin must be in its weak form. For a feminine singular noun in the nominative case, the weak adjective ending is -a.

If tignarlega is weak, why is the second adjective falleg in its strong form?

Falleg is a predicate adjective, describing the subject after the verb er (is). Because it doesn't directly follow the definite article Hin in a noun phrase, it uses the strong feminine singular nominative form, which is just the base word falleg.