1. The idea was that the West should be made safe for livestock
2. Ecologists recognize the important role that was once played by these predators.
3. Reintroducing a keystone species is one way that helps sustain the biodiversity.
4. In 1978 the USFWS proposed that gray wolves should be reintroduced.
5. What the suggestion did was bringing angry protests.
6. Some ranchers feared the wolves would attack their cattle.
The underlined clauses are the subordinate clauses in above sentences. A subordinate clause is a sentence-like structure with a subject, a verb, an object (if the verb is transitive), and adverb phrases, but they are instead a part of a sentence. They are also called dependent clauses.
In the first sentence the subordinate clause is the object of the sentence. In the second one, it defines the object "the important role," just like in the third one. In the fourth and sixth sentences, they are again the objects of the sentences. In the fifth sentence, the clause is the subject of the sentence.
There are 3 types of subordinate clauses: adverb clauses (time clauses, reason clauses, contrasting clauses, condition clauses), adjective clauses, and noun clauses.
I hope this helps!