Everyone has effectively listed the rules of usage for the semi-colon (which happens to be my favorite piece of punctuation), so I'll just add this observation. People who use semi-colons effectively are generally more mature writers. To use them correctly, one must not only understand the grammar rules (not all that easy, as demonstrated by the posts above) but also be able to construct a mature sentence. It's one of the standards, for me, of whether or not a student is making progress in writing.
Please keep in mind that it is okay to use a semicolon between two independent clauses, but those clauses must be closely related. Such as: My mother made homemade icecream for my birthday; she knows it’s my favorite. It would not be appropriate to say: My mother made homemade icecream for my birthday; it rained outside.
In response to post no.1 (first part):
A semicolon is used to join two compound clauses which are not joined by a coordinate conjunction. In this case it would replace the coordinate conjunction along with its preceding comma.
A semicolon can also be used to join compound clauses joined by a coordinate conjunction when there are already commas used within one or both of these clauses:
He had no further intercourse with spirits, but lived upon the Total Abstinence Principle, ever afterwards; and it was always said of him, that he knew how to keep Christmas well, if any man alive possessed the knowledge.
- from A Christmas Carol
This is similar to semicolons substituting commas for members of a series with internal clauses. In both cases it is used to differenciate between the first use and the second.
In response to post no. 3:
Semicolons are used instead of commas to separate members of a series which already have internal commas.
Charles Dickens was an avid user of this, and this was a popular structure in the latter 19th century:
All as they should be. Nobody under the table, nobody under the sofa; a small fire in the grate; spoon and basin ready; and a little saucepan of gruel upon the hob.
And every man on board, waking or sleeping, good or bad, had had a kinder word for another on that day than on any day in the year; and had shared to some extent in its festivities; and had remembered those he cared for at a distance, and had known that they delighted to remember him.
- from A Christmas Carol
Today syntax is more streamlined, and the tendency is to break up the idea into several distinct sentences instead.
Semicolons can also be used in a complex sentence that has three independent clauses. For example:
independent clause, and independent clause; independent clause. OR The dog barked, and I ran as fast as I could; however, the dog was faster and bit me.
My students never use semicolons. The only punctuation marks they are brave enough to use are commas, periods, and quotation marks.
Probably the most difficult of these uses for students to grasp is the use of semicolons to separate series of items which have internal punctuation. My students seem to know they can use a semicolon (;--the "period/comma" some of them call it...a "yield," not a complete stop but more than just a pause) to join two related sentences.
The Chicago Manual of Style's guidelines for the use of the semicolon can be found in sections 6.57-6.62.
The most common usage for a semicolon is between two independent clauses not joined by and, but, or some other conjunction (known as contact clauses). Example: Sarah went to the computer store; she didn't buy anything.
The two other most common usages include:
* before some adverbs (such as however, thus, accordingly) when they are used transitionally between independent clauses (example: James was very sick on Monday; however, he attended his classes.)
* in a series to avoid confusion when the items in the series have internal punctuation (example: The Johnson family took their dogs, Crusty and Bozo; their cats, Sneezy, Bashful, and Doc; and their birds, Tweeter and Woofer, to the veterinarian.)
Keep in mind that semicolons are stronger than a comma but weaker than a period. Writers sometimes use them for for stylistic reasons, such as to create a desired pause. The AP Stylebook's guidelines support this use, recommending that writers "use the semicolon to indicate a greater separation of thought and information than a comma can convey but less than the separation of a period implies." (p. 336) It's best to avoid the overuse of semicolons; the more you use them, the less impact they have.
I don't have the most recent edition of the MLA Handbook, so perhaps someone else can provide the MLA guidelines.