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Many of the concepts covered require the reader to abandon comfortable modes of thought
In the above sentence, the word "covered" is a verb. I was wondering could that sentence be rephrased like this - so that the word "covered" becomes an adjective:
Many of the covered concepts require the reader to abandon comfortable modes of thought
Yes, indeed, you may do this.
The form with -ed here is called the past participle, which means a form which participates in multiple contexts: it may act as a verb AND an adjective, or as a verb AND a noun, at the same time.
And in fact in you first sentence it is also acting as an adjective. Participles are often placed after the nouns they modify because they may, as verbs, have complements or adjuncts which in English may not come before the noun:
Many of the concepts [covered by this text] require the reader ... but not
✲ Many of the [covered by this text] concepts require the reader ...
The starred example is not permitted in English, though it is in at least one other closely related language, German. (Note that related there is another adjectival participle.)
✲ marks a usage as unacceptable
