Display
So you want to know what a display is huh? (If not; too bad I'm telling you anyway) a display is how you know that what you are doing is having an effect. A computer may still work and do everything you command with the display being either non existent or off, but you won't be able to tell if you are doing the right things, or if you are executing the commands that you intend to execute. You could turn your monitor off on a computer and type something out (if you felt like it) but you wouldn't know for sure if you had made a mistake, or if you had selected the correct area before typing and so on. This is why, even though the computer can function without the display being on, you wouldn't want to turn the display off.
Built in
A built in display is something that you would usually see on a portable device, such as a portable gaming device, or a laptop or tablet. With the display being built in this makes it more easily portable, because if the screen was a separate part of the device as the controller then it would not be very successfully portable, I mean look at the Virtual Boy.
That was supposed to be portable, but it was so not portable. It was a gigantic flop, and I believe that the controller being sort of separate to the display was just one of the reasons for this. There are many examples of the built in screen working well with portable computers, gaming and otherwise -
However, built in screens may not be the best choice for if you just want to use a computer at home, once again gaming or otherwise; if your screen is built in to the device you're using, it may force you to take an uncomfortable position in order to see it, whereas if your screen is separate to your device, you can take any position you want while the display stays wherever you put it, ready for you to look at it.
External
An external display has its benefits and it has its downsides. On the upside you don't have to keep your device near to you while watching in order to execute commands. But on the downside, you can't freely take a device with an external display anywhere you want and use it. With the display being its own separate thing, which allows you to upgrade the device and the display independent of each other.
Liquid Crystal
Liquid crystals are called liquid crystals because their molecules stay in the same sort of position in relation to each other, like with solid objects, but they can also move around like in a liquid. the molecules in liquid crystals aren't as free as those in normal liquids, but they aren't as uniform as those in normal solids either, which is why it's name suggest's that a liquid crystal is both solid and liquid. Liquid crystals react to electrical current - a specific kind of liquid crystal - twisted nematic liquid crystals, and they untwist variously depending on how much electrical current is flowing through them which is how they control how light passes through them.
Light Emitting Diode
Displays that use light emitting diodes are called a LED screens, they aren't really that different to LCD screens, they are pretty much the same but LED screens use LED's to light up the screen.
Organic Light Emitting Diode
OLED displays might seem like they aren't much different to LED displays, but they are quite different. There is no backlight and no twisting involved in an OLED screen OLED displays are self illuminating, they have the darkest blacks, they can make more colours, and they are more efficient. OLED screens are more efficient because when the screen shows black, that part of the screen is just not being illuminated, it turns off, showing the deepest blacks possible.
OLED screens are made of organic materials, and as explained they are more efficient than other kinds of screens, they are also easier to see from the side, the colours don't fade out as much as other screens when you look at it from the side, so why doesn't everyone have an OLED screen for everything? They look better because they are able to show more colours, and have higher contrast rates because of the whole deep blacks thing. Why wouldn't we just stop getting other kinds of screens? Well, OLED screens are typically more expensive than other kinds of screens, even though they are more efficient and won't take as much electricity to run as another screen. I can't really find another real reason not to buy one of these screens for everything.