What is the difference between a compound sentence and a complex sentence?

Just to add
a little more detail about compound sentences, there may be two or more
independent clauses:

"I was very thirsty, and I wanted
something refreshing; the lemonade stand a block down the street seemed more and more
enticing."

In this case, there are three independent clauses.
 Two of them are separated by the comma and the conjunction and.  The third
is set off by the semicolon, which can also be used to separate independent clauses.

Variety is important, and good writers learn to manage their sentence structure using
any combination/number of independent and dependent clauses.  Just be sure to use the tools to
separate them correctly, and strive for clarity.