Complex sentence structure

Let's start with simple sentences. A simple sentence is an independent clause that contains a subject and a verb.  On the forum, you've also seen discussion of compound sentences -- two independent clauses joined with a comma + conjunction such as "and" or "but" (there are other conjunctions as well).

A complex sentence has an independent clause joined by one or more dependent clauses. A complex sentence always has a subordinator such as because, since, after, although, or when or a relative pronoun such as that, who, or which. Sentences containing adjective clauses (or dependent clauses) are also complex because they contain an independent clause and a dependent clause. In the examples, the dependent clause is in bold.

The teacher returned the homework after she noticed the error.

After they finished studying, Juan and Maria went to the movies.

Although the definition of a complex sentence does not require it, complex sentence often begin with a subordinate/dependent clause.

When he handed in his homework, he forgot to give the teacher the last page.