My explanation comes from my teacher's edition from which I teach 4th grade Language Arts and from my own knowledge as an English teacher. Hopefully it is broken down simply and to the point.
A complex sentence contains two related ideas joined by a conjunction other than "and", "but", or "or". You can combine two short sentences to form a complex sentence.
Joe is busy. He has a job.
Joe is busy because he has a job.
Joe starts work. The sun rises.
Joe starts work when the sun rises.
Conjunctions that can be used:
when, as, as if, although, before, because, as though, if, after, since, where, though, while, until, wherever, unless
These are just a few examples.
A simple sentence has a subject, verb, and one complete thought. There are no conjunctions in a simple sentence. You might have a comma in a simple sentence, but a comma does not denote a compound or complex sentence. The conjunction is the key to recognizing a complex sentence. From there, you can find the independent and dependent clauses. Think about what the words independent and dependent mean. One clause is dependent on the other clause. That would be the dependent clause. The other clause can stand by itself. That would be the independent clause.
From there you can determine if the sentence if complex or not by using the tools I first explained above.
A simple sentence has one independent clause and no subordinate clauses. For example:
After the announcement of the final score, the local fans cheered loudly.
In this sentence there is a phrase ( After the announcement of the final score ) but only one independent clause ( the local fans cheered loudly.)
A complex sentence contains one independent clause and at least one subordinate clause. For example:
While we were in San Diego, we visited the Aerospace Museum, which is dedicated to the history of flight.
The independent clause is "we visited the Aerospace Museum". The subordinate clauses are "While we were in San Diego" and "which is devoted to the history of flight."
A simple sentence is a sentence that has a verb, a subject, and a complete thought. It is also known as an independent clause because even though i can be part of another sentence, it can still be a good sentence if it is alone. An example of a simple sentence would be: "Gordon ran around the park." Gordon would be the noun and ran would be the verb.
A complex sentence is a sentence that combines an independent clause with a dependent clause. A dependent clause is like an independent clause but it doesn't have everything it needs to make a complete sentence. An example would be: " Ivy and I found our friends at the pool, when we arrived late." The independent clause would be " Ivy and I found our friends at the pool..." and the dependent clause would be " ... when we arrived late.".
Simple sentences contain a subject and a verb and express a complete thought. For example: Rabbit loves gardening.
Compound sentences are just two independent clauses joined by a coordinator. For example: Pooh Bear loves to eat honey, but Rabbit does not have any honey to share.
A complex sentence contains and independent clause joined by dependent clauses. For example: Pooh Bear got stuck in Rabbit's door after he ate too much honey.