Inserting quotations (putting other people's words into your essay), and referring to other people's arguments, are both part of essay writing. (For referencing your quotations, see the Evidence - footnotes, endnotes, embedded references and bibliographies section of the site.)
Different types of quote:
In essays or assignments, you might quote somebody's exact words. For example: It was Bill Gates who remarked '640k ought to be enough for anybody'.
or you might take somebody else's argument and paraphrase their words. For example: Bill Gates always said that 640k should be enough memory for anybody.
Both these quotations need referencing, i.e. you must say where you found them. If you don't, you might well be accused of plagiarism, i.e. trying to pass off somebody else's words and work as your own.
Quotations are excellent when used properly, but should not be overused. Well chosen quotations add weight to an essay. Poorly chosen ones, particularly long ones that are clearly being used to pad an essay out, reduce an essay's value.
Quotation style
Quotations should NEVER be italicised (unless italicised in the original source, in which case the word [sic] can be used to show that these were the original author's italics). Nor should quotations be set out in a different font from the rest of the essay. Quotations should feel embedded into the flow of the essay, not set apart from it.
Quotations fewer than forty words should be kept within the text. For example: When Tony Blair said 'education, education, education'(1) he seems to have intended computers to be part of that education.
Quotations that are embedded in the text must be enclosed in single quotation marks.
You only use double quotation marks for a quotation within a quotation, for example: The Times commented that 'Lord Phillips, speaking in the House of Lords, declared "nobody in this house fully understands all of RIPA's intricacies" so we can hardly be expected to do better'.(2)
Quotations of over forty words (roughly 3 lines) should appear in an indented paragraph of their own, with a line space above and below. Indented quotations are not enclosed in quotation marks. Indenting is enough.
Removing or inserting words in quotations:
If you want to remove words from quotations - perhaps because the middle bit of the quotation is a long and unnecessary list - you must show you have done this with ... (ellipses).
For example: One commentator suggested that 'children should not be microchipped ... under any circumstances'. Ellipses are not necessary at the beginning or end of a quotation.
If you have to insert a word because something doesn't quite make sense, you show that with square brackets. For example: Nobody paid much attention to Twitter until Stephen Fry told the world he was stuck in a lift. As one commentator wryly noted 'he [Stephen Fry] suddenly made tweeting not just for twits'.
Where to put the full stop:
If a quotation finishes at the end of a sentence, the reference number goes after the full stop. For example: Tony Blair finished his speech repeating 'education, education, education'.(3)
If the quotation is set within the text, the reference number follows the closing quotation mark. For example: Tony Blair said 'education, education, education'(4) but what he classed as education was unclear.
Whole sentence quotations:
Only on the very very rare occasions when a quotation is a whole sentence should the full stop be placed before the closing quotation mark. For example: 'I believe in education, education, education.' It's not good practice, though, to simply stick in a whole sentence as a quotation. Quotations need to be introduced, and need to make a point. For example: When Tony Blair said 'I believe in education, education, education'(1) he was simply looking for a soundbite.