A complex sentence contains at least one main (independent) clause and one subordinate (dependent) clause:

Time runs quickly when a deadline is near.

Since there was no food, he was hungry.

The phrases "when a deadline is near" and "since there was no food" are subordinate clauses. The words 'when' and 'since' are subordinating conjunctions.

One test for a subordinate clause is to see whether the clause could stand as a separate sentence. The clause is subordinate if it does not express a complete thought on its own. Often the clause is made to be subordinate by the subordinating conjunction and could stand on its own with the conjunction removed.