A compound-complex sentence consists of two or more coordinate clauses and at least one subordinate clause.

Consider the sentence given below:

Macaulay had wealth and fame, rank and power and yet he tells in his biography that he owed the happiest hours of his life to books.

A: Principal clause: Macaulay had wealth and fame, rank and power
B: Principal clause: And yet he tells us in his biography
C: Subordinate noun clause: That he owed the happiest hours of his life to books (Object of tells in B)

Now consider the sentence given below:

A gentleman never speaks of himself except when compelled, never defends himself by a mere retort and he has no ears for slander or gossip.

A.    Principal clause: A gentleman never speaks of himself
B.    Subordinate clause: Except when compelled (Adverb clause of time modifying the verb ‘never speaks’)
C.    Principal Clause: Never defends himself by a mere retort (Coordinate with A)
D.    Principal Clause: And he has no ears for slander or gossip (Coordinate with A)

While the man played the guitar, the boys sang and the girls danced.

A.    Principal clause: The boys sang
B.    Principal clause: The girls danced (Coordinate with A)
C.    Subordinate clause: While the man played the guitar (Adverb clause of time)

Shortened forms of the compound sentence

Omission of the conjunction

We sometimes omit the conjunction in a compound sentence.

I came, I saw, I conquered. (OR I came, I saw and I conquered.)
I laugh, I run, I leap, I sing for joy. (OR I laugh, I run, I leap and I sing for joy.)

Omission of the subject

When the clauses have a common subject, it is usually omitted in the second and subsequent clauses.

The horse neighed and pawed the ground. (More natural than ‘The horse neighed and the horse pawed the ground.)
He wrote the letter, sealed it and posted it. (More natural than ‘He wrote the letter, he sealed it, and he posted it.)

Omission of the predicate (when the clauses have a common predicate)

One man threw a tomato and another an egg. (OR One man threw a tomato and another threw an egg.)
Some played hockey and others cricket. (OR Some played hockey and others played cricket.)