Quotation marks are “, the left quotation mark, and ”, the right quotation mark. It should be noted that not all cultures use these this way. For a good explanation on various language and culture difficulties, as well as quotation marks in many other cultures, see avjewe's writeup below.
Unicode and HTML entities generally help. This writeup discusses some ASCII limitations.
In ASCII-based communications, your ordinary quotation marks are usually ASCII 34 ("), and single quotes are ASCII 39 ('). This is usually all you need to know if you are storing information on computer to be processed by computer and to be displayed on computer.
These symbols are enough for writers behind the keyboard, but not for printers who put the thing on paper.
For typographical reasons, ASCII 34 usually represents an inch symbol and ASCII 39 is an apostrophe. The actual quote marks don't have their representation in ASCII (and not even in Latin-1, though some "extended" ASCIIs have them).
TeX notation, used by some people in ASCII, uses ``this type of of quote marks'' (Repeated ASCII 96 and 39). This is a good way but somewhat space-wasting in ASCII text - originally, TeX processed these to single characters. Likewise, outside USA, in English, things can be quoted with `single quotes', with single marks, obviously. We here in Finland can quote phrases with ''double quotes'' and 'single quotes', with just madly banging the apostrophe key.
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A small request for the readers: if you are doing DTP work, try to get the quotes right! It is one of the problems that is most obvious to spot, the single most obvious mistake people make when they first use DTP applications and prepare something to be published on print. It has been a long time since the so-called DTP revolution and I still see people using "straight quotes" or “smart quotes” when they should be using ”correct quotes for Finnish”. Some even use “wacky quotes“. Please refer to the style guides and recommendations and do what professionals do - it isn't that difficult. Small bits of caring go a long way.