A semicolon (;) is the punctuation mark used to connect independent clauses, indicating a closer relationship than does a period.
The rules are pretty simple, though they can be confusing to some. As the above definition shows, semicolons are used inside one sentence, to connect closely related independent clauses. The 1983 Guinness Book of World Records names the 1,288 word sentence in William Faulkner’s Absolom, Absolom as “the longest sentence in literature”.
The most common misuse I’ve seen is people using them to connect totally unrelated clauses or sentences. Something like this would be wrong:
The cat ate a mouse; a car turned in the driveway.
A correct example is the following sentence:
Some people write with a word processor; others write with a pen or pencil.
(example from The Writing Center @ The University of Wisconsin)
Semicolons are used to separate items in a series, such as:
This conference has people who have come from Boise, Idaho; Los Angeles, California; and Nashville, Tennessee.
(example from GrammarBook.com)
Here are some more links that give examples of correct usage: