The bracketed expression is an infinitival relative clause that modifies the noun "book". "Read" is a transitive verb, so it requires an object, but the object is missing, so we represent it with a ___ (called "gap"). In this case the missing noun object of "read" is clearly "book": "to read a book". It's easy to tell that it's an infinitival relative because it has a meaning close to the finite relative clause "I need a book that I can read on holiday".
Some people call relative clauses "adjective clauses" because they modify nouns, but that's not really correct since they are not headed by an adjective but by a verb.
2) "He opened the box [to reveal a huge bullfrog]".
This time the bracketed expression doesn't modify a noun, but the verb phrase "opened the box". It may look as though it modifies "box", but it doesn't because noun modifiers help to identify or describe the noun; this clause explains what happened when he opened the box. In other words it explains the result (or possibly the purpose) of opening it. Thus it is a result - or purpose - adjunct (adverbial).