What is an LCD?
LCD stands for Liquid Crystal Display. An LCD is a passive device, i.e. it doesn't actually produce any light, it simply alters the light travelling through it. With a voltage applied to it the liquid crystal polarises transmitted light in a different direction to when no voltage is applied (explanation here). A polarising filter in front of the display then blocks one of the two polarisations (i.e. the areas in which a voltage was applied or the ones where no voltage was applied) and therefore in some areas of the screen appear 'black', this effect is used to draw the characters and graphics displayed by an LCD.
Almost all LCD's have a strong light source built in behind a glass panel (which contains the liquid crystal), this ensures that the areas of light and dark on the screen (dictated by the areas on the liquid crystal panel across which a voltage is applied) have good contrast. Displays with no backlights that rely solely on the light incident on the LCD panel cannot be used in low light conditions.
What is a VFD?
VFD stands for Vacuum Florescent Display. Unlike an LCD a VFD is an active device; it makes its own light. It does this in exactly the same way as a CRT monitor or television, by accelerating electrons at a phosphor target from a hot element. A very well written guide explaining how this principle is applied to make a whole display can be found here.
What are they used for?
In general they are used to relieve some of the displaying load from your monitor, showing everything from system stats to current song playing and game stats. They can also be used as monitor replacements in mp3 boxes for example.
There are two main ways of interfacing an LCD with your computer Serial and Parallel. A serial display is normally a parallel display with a serial controller bolted on...
Alphanumeric and graphic?
An alphanumeric display can only display characters (normally in 5x7 pixel blocks at specific locations on the screen). Screen size is measured by the number of characters the display can fit on one line and the number of lines in can display, e.g. 20x2 (20 characters per line and 2 lines). 16x2, 20x2, 24x2, 20x4, 4x20 and 4x40 are the most common sizes available.
Graphic displays consist of an array of pixels which can be controlled at whim to display anything. 5x7 is the 'normal' font size; hence a 122x32 display is the graphic equivalent of a character 20x4.
Serial and parallel LCD's you say, so what's the difference?
Serial displays (RS232, SPI, TTL, USB, I2c,RS485... - with a little bit of wiring you can interface all of the following with an RS232 or USB port), they generally enjoy the best software support, are easy to interface with, allow for long data cables. The disadvantage of serial controllers is the interface speed, which isn't enough for full screen bit-map animations at any reasonable resolution.
For character only (alphanumeric) displays the following serial controllers are the most common:
-Matrix Orbital (RS232/I2C). There are quite a few different MO serial controllers in general they have; decent speeds (19.2Kbps is standard - which is about right for a 4x40 or smaller display), keypad support (allows momentary switches/keypad units to be attached to the LCD to control software (e.g. WinAmp), GPO's (General Purpose Outputs - basically driver controlled switches, so you can use you LCD to control your case fans or lights), software contrast control (useful for fading into screens, but a little pointless generally), software backlight control, 8 custom chars, custom boot screens on some displays, wide voltage input as an option (due to a silly power connector design normal voltage displays are easy to destroy). An LCD is only as good as it's software support and thankfully software support for all Matrix Orbital LCD's is good, support software is listed later on - try the software out before you buy! Buy from 
www.matrixorbital.com or 
http://www.hvwtech.com/ for a copy-cat controller.
-Crystalfontz (RS232). A few diff types of controller here, their newest (the 633) is impressive but needs more software support. In general CF series boards (v2.0) are capable of, good speeds (up to 19.2Kbps), custom characters, contrast memory, large character sets, semi-graphic support for smooth scrolling and probably some small features I've missed (the 633 can PWM control fans and have temperature sensors added to it!). On the software front the CF display fairs well, some major pieces of LCD software work with this controller. Buy from 
www.crystalfontz.com.
-Seetron (RS232) (Scott Edwards Electronics). The controllers vary depending on LCD size but in general they're very primitive. They feature below average speeds (9.6Kbps max I think, fine for 2x20 or smaller), standard character sets and on board contrast control (nice for permanently setting the contrast). The seetron controller can be made to work with LCDCenter, Driver and maybe LCDC - though functions are limited. Buy from 
http://www.milinst.com/ or 
http://www.seetron.com/.
-Wirz (RS232). Very like Seetron, buy from 
http://www.wirz.com/
-Others (inc. in-built controllers). Most other serial controllers are fairly simple (and are often very similar to Seetron's). Exceptions to this rule are inbuilt controllers, like the one made by Noritake which is feature packed (in capable of great speeds and have 8 I/O's which can be used in the same way as GPO's or keypad interfaces).
For character only (alphanumeric) displays 99% with parallel interfaces follow a single standard. They're nearly all HD44780 compatible (for 4x20 character displays or smaller). With parallel displays almost everything is done at the PC end which makes development more difficult, though theoretically you can do anything with a parallel LCD. They are very fast, with tops speeds of 115Kbps and normally have an inbuilt character set allowing for 8 custom characters. Software support is hopeless if you choose a display that doesn't use a HD44780 or compatible controller, but thankfully most do. 
For graphic displays the controller situation is a little different. Quite a few have serial interfaces which makes controlling them easy, but rules out full screen bitmap animation. On the parallel front the controllers vary wildly though we're starting to see some support appear. A closer look at some of the options:
-Noritake (RS232, SPI, async, TTL and parallel). Their serial controller is pretty advanced and very easy to use, more info in this review - there's good support for this controller. Their parallel controller is also very nice, though software support for it isn't great (see later). Find Noritake here.
-Matrix Orbital (RS232, I2C). Matrix Orbital do a range of graphical displays which have a similar feature list to their alphanumeric displays. They have font and image memory and support features like hardware scrolling. Find Matrix Orbital here.
-Toshiba's T6963C controller and other large graphic controllers (parallel). There is now beginning to be some software written for some of the larger graphicals. The Toshiba T6963C controller is commonly seen on 240x64 and 240x128 displays and a couple programs will support this chipset (see later). These are relatively cheap displays, and very nice they are but they're nearly all green). Before you buy a large graphic display make sure you know how to wire it up and that it's controller has some software support, if it doesn't the display you've bought will be as useful as a chocolate fire guard.