If you're like me, you've been looking around the internet for a good tutorial on using the ESP8266 with neopixel LEDs (AKA the WS2812 or WS2812b), programmed via the Arduino IDE. I haven't found it all in one place, in English, but it turns out it's pretty easy now - much better than a few months ago. First, props to sabas1080 for his instructable, which does a nice job of this in Spanish. I'm documenting it here in English in case this helps others.
This is a basic breadboard demonstration. It's not meant to be a "final product" but consolidates instructions for this basic setup. If you haven't used the ESP8266 yet, it's an awesome little wifi board that you can get for a few bucks. Even the fancy version is only $10.
Note that this DOES NOT require a separate Arduino board, it's running the neopixels directly from the ESP8266 and just programming it with the Arduino IDE. And thanks to the hard work of many people it's now very easy!
Parts you'll need:
Note: using the Huzzah breakout board makes things easier because it has buttons built in for reset and bootloader mode. if you're using a bare ESP8266 board, you'll want a couple additional buttons to use for reset and putting the ESP8266 in the bootloader mode (to upload your code to it).
Assemble as shown in the diagram above. The capacitor and resistor help protect your neopixels. I'm using a neopixel ring because that's what I happen to be working on, but you can test with a strip or a single pixel or whatever you have. If you are using the Adafruit Huzzah breakout, solder it as shown on their nice page here: https://learn.adafruit.com/adafruit-huzzah-esp8266-breakout/using-arduino-ide using your breadboard.
You'll need the FTDI cable or USB for programming. After the programming is finished, you can remove it. Speaking of which, let's program this thing.