COMPLEX SENTENCE

COMPLEX SENTENCE
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. In the following complex sentences, subjects are in yellow, verbs are in green, and the subordinators and their commas (when required) are in red.

A. When he handed in his homework, he forgot to give the teacher the last page.  B. The teacher returned the homework after she noticed the error. C. The students are studying because they have a test tomorrow. D. After they finished studying, Juan and Maria went to the movies. E. Juan and Maria went to the movies after they finished studying.

When a complex sentence begins with a subordinator such as sentences A and D, a comma is required at the end of the dependent clause. When the independent clause begins the sentence with subordinators in the middle as in sentences B, C, and E, no comma is required. If a comma is placed before the subordinators in sentences B, C, and E, it is wrong.
Note that sentences D and E are the same except sentence D begins with the dependent clause which is followed by a comma, and sentence E begins with the independent clause which contains no comma.  The comma after the dependent clause in sentence D is required, and experienced listeners of English will often hear a slight pause there.  In sentence E, however, there will be no pause when the independent clause begins the sentence. 
COMPLEX SENTENCES / ADJECTIVE CLAUSES
Finally, sentences containing adjective clauses (or dependent clauses) are also complex because they contain an independent clause and a dependent clause.  The subjects, verbs, and subordinators are marked the same as in the previous sentences, and in these sentences, the independent clauses are also underlined. 

A. The woman who called my mom sells cosmetics. B. The book that Jonathan read is on the shelf. C. The house which Abraham  Lincoln was born in is still standing. D. The town where I grew up is in the United States.

Adjective Clauses are studied in this site separately, but for now it is important to know that sentences containing adjective clauses are complex.