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!