In this example, the sentence:
"After he played all day the little boy sleepy and contented"
is a fragment. This is not a complete sentence because it lacks a verb. However, if we were to rewrite the sentence with the addition of a verb:
"After he played all day the little boy was sleepy and contented"
then the sentence would be a complex sentence.
Compound sentences refer to those that are made up of two independent clauses that could stand on their own independently. On the other hand, complex sentences are those that are formed by an independent and a dependent clause:
After he played all day = dependent clause
the little boy was sleepy and contented = independent clause