OLED, or Organic Light Emitting Diodes, are the best advancement in display tech since LCD itself. Their inky blacks lead to image quality that outpaces traditional LCD, and unlike LCD-based displays they don't require a backlight, so they can be super-thin.
The advantages of OLED show up screens both tiny and huge.
Many of the the flagship smartphones currently use OLED displays. Samsung is the major producer of small screens OLED for phones, found not only in Samsung Galaxy phones but also in the Apple iPhone X, Google Pixel 2 and the OnePlus 5T.
CNET's picks for the best-performing TVs are OLED as well. LG Display is the only company currently producing big OLED screens TVs, found in LG TVs as well OLEDs by Sony and, outside the US, Panasonic and Philips.
LG also makes phone-size OLED screens, most notably in the the Pixel 2 XL, but that phone showed issues like burn-in and worse image quality that didn't affect Samsung-made OLED phone screens. On the TV side, LG's TV OLEDs are widely lauded, while Samsung hasn't sold an OLED TV since 2013.
Isn't OLED just OLED? Yes and no. Though the technology and methods are similar, the materials, how they're made, and so on can result in significant differences.
Both Samsung and LG talk about AMOLED and POLED, both of which are ways to describe different parts of an OLED display.
"AM" stands for "active matrix," which describes how the individual OLED pixels are addressed. An OLED with a passive matrix would be fine for a fitness tracker, but anything you want to watch video on needs active matrix. Which is to say, as far as phones and TVs go, all OLEDs are active matrix. Samsung calling their OLEDs "AMOLED" is as redundant as if they called their LCDs "AMLCD" since they're also all active matrix.
The "P" in LG's marketing stands for "plastic," as opposed to glass. This refers to the base material of the OLED display (the "substrate"), not the front part you touch -- which is separate and often Gorilla Glass. Plastic is lighter and works better for phones, and allows the display to be more easily curved, which is why both Samsung and LG use it.
The main thing to know is that while not all POLEDs have to be AMOLEDs or vice versa, these days most OLED displays are both.
Or to put it another way, both Samsung and LG could call their current OLED phone displays OLED, POLED, AMOLED, or even PAMOLED, the last one to the delight of Pams everywhere, I'm sure.