When are semicolons correctly and effectively used?

There are two main ways to use semicolons.

1. To Join Two Independent Clauses in a Sentence

The first is to separate what are technically called independent clauses. Just think of these as complete sentences—they have a subject and a verb, and they could stand alone as sentences of their own. Take these two sentences, for instance:

• Jeff is tall.
• Bill is short.

These are complete sentences, and to join them together into one sentence, we have to use a semicolon:

• Jeff is tall; Bill is short.

We cannot use ONLY a comma, as this would create what's called a comma splice. (We could use a comma AND a conjunction like and or but, but not just a comma by itself.)

• WRONG: Jeff is tall, Bill is short 
• RIGHT: Jeff is tall, but Bill is short.
• RIGHT: Jeff is tall, and Bill is short.
• RIGHT: Jeff is tall; Bill is short.

2. (Sometimes) to Separate Items in a List

The second way to use semicolons is to separate items in a list. Normally we just do this we commas, like this:

• There are three types of tree in the orchard: apple, fig, and pear.

But sometimes commas can cause confusion in a list, especially if some of the items within that list have commas of their own. For instance:

• Some of the world's most popular cities to visit are Paris, France, Rome, Italy, and London, England.

Here, it looks like my list contains six items (Paris AND France AND Rome AND Italy AND London AND England.) But really it contains only three items (Paris, France AND Rome, Italy AND London, England), and to clarify that fact I can use semicolons:

• Some of the world's most popular cities to visit are Paris, France; Rome, Italy; and London, England.

Here, it's clear that I mean to have three cities, and not six places, in my list.