1. Semi-colons are used to join two RELATED independent clauses.
Gem had been running all day. She was very tired.
These independent clauses are closely related, so we can join them with a semi-colon. Note that we do not capitalize the first letter of the second clause (unless it's a proper noun, that is).
Gem had been running all day; she was very tired.
Do not use a semi-colon to join two clauses if they are not closely related.
Drannor had been running all day. Gem liked to eat peaches for lunch.
Eragon had been running all day; Arya liked to eat peaches for lunch. [WRONG USE] These sentences should NEVER be joined with a semi-colon. I've exaggerated the point here by choosing two sentences that are completely unrelated, but you can see how the first example is different than this one.
2. Semi-colons are used to join items in a series.
When we have a simple series, we join the items with commas.
Drannor ate peaches, berries, and nuts for dinner.
When the items in the series already have commas, we use semi-colons to separate them.
Gemma was escorted by Drannor, a Rider; Gem, a princess; and Gaelira, a dragon.
There are actually several times a colon should be used, but we'll restrict ourselves to the one you're more likely to use.
Use a colon after a complete statement or independent clause to introduce a list, a quotation, or some other type of clarifying statement.
Drannor learned many things: swordplay, archery, and patience.