Argthtjtr

I'm quite unsure regarding the usage of single quotation marks (') and double quotation marks (") in English.

I had thought that double quotation marks were usually used to quote sentences from passages/given sources, nouns/things ("Westminster Bridge", "alliteration", or "voice" regarding its usage in poetry), as well as some less common/important uses including being snarky and using them to indicate a sarcastic remark.

Someone had told me today that you were supposed to refer to things with two inverted commas (") instead of one.

Which one is correct? Could someone explain the usages between the two different types of quotation marks?

This is simply a question of style. Wikipedia has a huge article on the subject. The passages most relevant to your question are:

Quotations and speech
Single or double quotation marks denote either speech or a quotation. Neither style—single or double—is an absolute rule, though double quotation marks are preferred in the United States, and both single and double quotation marks are used in the United Kingdom. A publisher’s or author’s style may take precedence over national general preferences. The important rule is that the style of opening and closing quotation marks must be matched[.]

[...]

Use–mention distinction

Either quotation marks or italic type can emphasize that an instance of a word refers to the word itself rather than its associated concept. [...] A three-way distinction is occasionally made between normal use of a word (no quotation marks), referring to the concept behind the word (single quotation marks), and the word itself (double quotation marks). [...] In common usage, there may be a distinction between the single and double quotation marks in this context; often, single quotation marks are used to embrace single characters, while double quotation marks enclose whole words or phrases[.]

Emphasis mine. Read the entire article for further insight.

According to the The Oxford Guide to Style British usage of single vs double inverted commas differs from the US one:

Quotation marks, also called 'inverted commas', are of two types: single and double. British practice is normally to enclose quoted matter between single quotation marks, and to use double quotation marks for a quotation within a quotation:

• 'Have you any idea', he said, 'what "dillygrout" is?'

This is the preferred OUP practice for academic books. The order is often reversed in newspapers, and uniformly in US practice:

• "Have you any idea," he said, "what 'dillygrout' is?"

If another quotation is nested within the second quotation, revert to the original mark, either single-double-single or double-single-double. When reproducing matter that has been previously set using forms of punctuation differing from house style, editors may in normal writing silently impose changes drawn from a small class of typographical conventions, such as replacing double quotation marks with single ones, standardizing foreign or antiquated constructions, and adjusting final punctuation order.

Do not, however, standardize spelling or other forms of punctuation, nor impose any silent changes in scholarly works concerned with recreating text precisely, such as facsimiles, bibliographic studies, or edited collections of writing or correspondence.

Josh's answer citing The Oxford Guide to Style (a British publication) neatly summarizes the difference between predominant UK and US English punctuation styles regarding double and single quotation marks. Beyond that various U.S. style guides offer similar analyses and some interesting sidelights on the predominant U.S. style.

From The Chicago Manual of Style, fifteenth edition (2003):

11.33 Quotations and "quotes within quotes." Quoted words, phrases, and sentences run into the text are enclosed in double quotation marks. Single quotation marks enclose quotations within quotations; double marks, quotations within these; and so on. ... The practice in other English-speaking countries is often the reverse: single marks are used first, then double, and so on.

...

11.35 Quotation marks in block quotations. Although material set off as a block quote is not enclosed in quotation marks, quoted matter within the block quote is enclosed in double quotation marks—in other words, treated as it would be in text (see 11.33). An author or editor who changes a run-in quotation to a block quotation must delete the opening and closing quotation marks and change [reverse] any internal ones. ... Similarly, converting a block quotation to a run-in quotation requires adding and altering quotation marks.

Chicago also has several entries for special uses of quotation marks:

7.52 Parentheses and quotation marks. A translation following a foreign word, phrase or title is enclosed in parentheses or quotation marks. ...In linguistics and phonetic studies a definition is often enclosed in single quotation marks with no intervening punctuation; any following punctuation is placed after the closing quotation mark.

...

7.58 "Scare quotes." Quotation marks are often used to alert readers that a term is used in a nonstandard, ironic, or other special sense. Nicknamed "scare quotes," they imply "This is not my term" or "This is not how the term is usually applied." Like any such device, scare quotes lose their force and irritate readers if overused. ... In works of philosophy, single quotation marks are sometimes used for similar purposes, but Chicago discourages that practice unless it is essential to the author's argument and not confusing to readers.

...

8.138 Horticultural cultivars. Many horticultural cultivars (cultivated varieties) have fanciful names that must be respected since they may be registered trademarks. Such names should be enclosed in single quotation marks; any following punctuation is placed after the closing quotation mark.If the English name follows the Latin name, there is no intervening punctuation.

In short, in the special cases of linguistics terms and botanical cultivars, Chicago (and general U.S.) usage follows British English conventions for quotation mark choice and for placement of other punctuation relative to the quotation marks.

The Associated Press Stylebook (2002) has far less to say about single quotation marks, preferring double quotation marks for every purpose except instances involving nested quotations:

QUOTES WITHIN QUOTES: Alternate between double quotation marks (“ or ”) and single marks (‘ or ’):

She said, "I quote from his letter, 'I agree with Kipling that "the female of the species is more deadly than the male," but the phenomenon is not an unchangeable law of nature,' a remark he did not explain."

Use three marks together if two quoted elements end at the same time: She said, :He told me, 'I love you.'" (NOTE: Local style should ensure the same differentiation between the single and double quotation marks, either with a "thin" space or by different typography, if not computer programmed.)

This passage—which echoes the advice in Chicago—is the only place in AP that mentions single quotation marks at all; in every other situation involving quotation marks, it is understood that double quotation marks are to be used.

Words into Type, third edition (1974) adds a few wrinkles to its discussion of single quotation marks, but in general its treatment is consistent with Chicago's and AP's:

Excerpts. All direct quotations from the work of another should be enclosed in [double] quotation marks unless they are set as extract, in type smaller than the text type, or set solid within a leaded text. ... A quotation within a quotation should be enclosed in single quotation marks. If this, in turn, contains a quotation, this last should be double quoted. One within this should be single quoted.

...

Single quotation marks. Single quotes are properly used to enclose quoted matter within a quotation [cross reference omitted]. Other uses are more or less arbitrary and unusual. They may be used to save space when quotations are numerous, or to differentiate certain terms from other similar.

29. dissilire: 'is split,' as we speak of "a splitting headache."

A once-common practice, now usually confined to etymological texts, was the use of double quotes only for extracts and dialogue and single quotes in all other cases where quotes were called for.

At the U.S. computer magazine where I worked for many years, we followed Chicago style in using double quotation marks in almost all situations involving quotation marks (other than quotations within quotations). But we broke with Chicago in using single quotation marks when quoting particular computer menu options (as opposed to computer message text, which we used double quotation marks to quote). For example:

Beneath the text block that reads "Other Valuable Benefits for Our Treasured Customers," uncheck the box next to 'Please keep me informed of other exciting products from XCorp.'

The idea was to help readers distinguish between computer text that involved simple reading and computer text that required some sort of choice or action on the user's part.

One use of single and double quotation marks not mentioned in any of the other answers is when quotation marks need to be nested, as in the case where a quotation itself includes a quotation. For example,

John stated, "I was walking down the sidewalk, when I heard a woman cry, 'Get out of here!'. I couldn't tell who she was talking to."

In this case, using double quotes both for John's statement, and for the exclamation of the woman would be confusing.

I have been using double marks when writing and have never used single marks. Those kinds of preferences are in the eye of the beholder. However, others might argue that this is a difference between British English and American English.

EDIT (23/9/14): For example, while U.S. English spells defence 'defense', British English spells it 'defence'.

Adding to the insight given by RedDwight, I found that in practice single marks are commonly used for single words or short sentences while double marks are used to denote longer passages of text. This may have become naturalized to some (me), but I don't know of any consensus on this.