The Elements of Style uses four different overlapping grammatical terms in the first part. It discusses relative clauses and restrictive vs. non-restrictive ones as well as appositives and participial phrases. It gives grammatical details; here I just want to give examples and describe how they stand in relation to one another because they were similar and I got confused.
Relative clauses are clauses that describe a noun. They are divided into restrictive and non-restrictive clauses. Restrictive clauses emphasize the description while nonrestrictive ones give additional, incidental information. A restrictive clause: "The man *who came in here last night* is Paul Dano." It's often introduced by who, whom, or that, but the introducing word is sometimes left off. A non-restrictive clause: "The drunk dude, *who once fought a whole legion of raccoons,* is down the road fighting a toad." There's generally a verb in
Appositives are when you put two noun phrases side by side. Like: *My sister* **Gloria** is a secret spy. "Gloria" is a noun phrase that further specifies or describes "My sister." This is also a restrictive modifier, but not a relative clause because "Gloria" is not a clause. "My sister, Gloria, is unpacking the bags" is a nonrestrictive example.
Participial phrases look like: *Pulling the jar from her bag,* Helen raced down the steps. Verbing, blah blah blah. The verbing describes the subject, in this case Helen.
The Elements of Style gives more information about each of these and how to keep the grammatical structure clear.