Complex: Contains an independent clause plus one or more dependent clauses. (A dependent clause starts with a subordinating conjunction. Examples: that, because, while, although, where, if.)
I don't like dogs that bark at me when I go past.
She did my homework, while her father cooked dinner.
You can write on paper, although a computer is better if you want to correct mistakes easily.
Note: A dependent clause standing alone without an independent clause is called a fragment sentence - see below.