Compound sentence

A compound sentence consists of two or more complete sentences (independent clauses) combined into one:

Toes are chopped off; severed fingers fly through the air.*
Toes are chopped off; [INDEPENDENT CLAUSE] severed fingers fly through the air. [INDEPENDENT CLAUSE]

Toes are chopped off. [COMPLETE SENTENCE/INDEPENDENT CLAUSE]
Fingers fly through the air. [COMPLETE SENTENCE/INDEPENDENT CLAUSE]

There are 4 ways to combine independent clauses:

How to join independent clauses (compound sentence)
Important: these methods do not apply to joining a dependent clause with an independent clause (complex sentence)
Comma and FANBOYS (for, and, nor, but, or, yet)   Toes are chopped off, and severed fingers fly through the air.
Semicolon   Toes are chopped off; severed fingers fly through the air.
Semicolon, “fancy FANBOYS,” & comma
(“fancy FANBOYS” = adverbial conjunctions or conjunctive adverbs)
Toes are chopped off; moreover, severed fingers fly through the air.
Colon The main reason that Zipes likes fairy tales, it seems, is that they provide hope: they tell us that we can create a more just world.*
“Don’ts”
DO NOT join two independent clauses with a comma! **   Toes are chopped off, severed fingers fly through the air. WRONG!
DO NOT simply run the two independent clauses together! ***   Toes are chopped off severed fingers fly through the air. WRONG!

* source:
Acocella, Joan. “The Lure of the Fairy Tale.” The New Yorker. 23 July 2012. Print.
** As with so many grammar and writing rules, the comma-splice rule has an exception. However, I wouldn’t worry about it for college writing. When you are writing papers for college classes, do not use a comma to combine independent clauses.
*** There are no exceptions to this rule that I’m aware of.

AND SEE:
5+2: the canonical sentence (clause) patterns
Independent clause
Compound sentence – how to punctuate
Complex sentence
3 ways to combine the 7 clause patterns

Commas after “introductory elements”

For example, [INTRODUCTORY PHRASE] the story of “Little Red Riding Hood” has appeared in many guises.

Reading “Little Red Riding Hood,” [INTRODUCTORY PARTICIPLE CLAUSE]* I am struck by the story’s lack of logic.

When I was little, [INTRODUCTORY ADVERBIAL CLAUSE] my mother read me the story of “Little Red Riding Hood.”

NEVER use a semicolon after an introductory element!

For example; the story of “Little Red Riding Hood” has appeared in many guises.
Reading “Little Red Riding Hood;” I am struck by the story’s lack of logic.

Doesn’t work!

* The participle clauses are often called the participle phrase.