A comma marks a slight break between different parts of a sentence. There are four common occasions on which commas are necessary; follow the links for more information.

Using commas in lists (e.g. The flag was red, white, and blue.)
Using commas in direct speech (e.g. 'That's not fair,' she said.)
Using commas to separate clauses (e.g. As we had already arrived, we were reluctant to wait.)
Using commas to mark off parts of a sentence (Her best friend, Eliza, sang for a living.)

The semicolon is most commonly used to mark a break that is stronger than a comma but not as final as a full stop. It’s used between two main clauses that balance each other and are too closely linked to be made into separate sentences. For example:

The film was a critical success; its lead actors were particularly praised.

Colon

There are three main uses of the colon:

between two main clauses in cases where the second clause explains or follows from the first:

We have a motto: live life to the full.

to introduce a list:

The cost of the room included the following: breakfast, dinner, and Wi-Fi.

before a quotation, and sometimes before direct speech:

The headline read: ‘Local Woman Saves Geese’.

Apostrophe

There are two main cases where apostrophes are used:

Using apostrophes to show possession

Malcolm's cat was extremely friendly.

We took a day trip to the Gibsons' house.

Using apostrophes to show omission

I'm afraid the pie isn't suitable for vegetarians.

We didn't think about the consequences of our actions.

Hyphen

Hyphens are used to link words and parts of words. There are three main cases where you should use them:

in compound words

My mother-in-law is visiting soon.

to join prefixes to other words

The novel is clearly intended to be a post-Marxist work.

to show word breaks

He collects eighteenth- and nineteenth-century vases.

Dash

There are two main occasions on which a dash can be used, usually in informal writing:

to mark off information that is not essential to an understanding of the rest of the sentence

Many birds—do you like birds?—can be seen outside the window.

to show other kinds of break in a sentence where a comma, semicolon, or colon would be traditionally used

Tommy can't wait for Christmas—he's very excited.

Brackets

There are two main types of brackets. Round brackets are mainly used to separate off information that isn’t essential to the meaning of the rest of the sentence: He asked Sarah (his great-aunt) for a loan. Square brackets are mainly used to enclose words added by someone other than the original writer or speaker, typically in order to clarify the situation: The witness said: 'Gary [Thompson] was not usually late for work.'