Usually after a complete sentence and...
1) Before a list.
I could only find three of the ingredients: sugar, flour and coconut.
2) Before a summary.
To summarise: we found the camp, set up our tent and then the bears attacked.
3) Before a quote.
As Jane Austen wrote: it is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife.
4) To complete a statement of fact, where the colon is used in place of the following or thus.
There are only three kinds of people: the good, the bad and the ugly.
semi colon
use them to link two complete sentences (using a comma is comma splicing)
I had to go home early; I had a head cold.
or linking two sentences witha conjunctive adverb
I would have worked late; however, I had a terrble head cold.
or in a list where commas are already present
On vacation this year we visited the following: Vienna, Austria; Zurich, Switzerland; and Siena, Italy.
