From what I understand, WS2801 and LPD8806 are both SPI-based chipsets. Both do 8-bit PWM and have RGB channels. Both come as separate chips which are connected to any sort of RGB LED, available as fairy-light strings or as strips. LPD8806 seems to be patent encumbered and less available for purchase than the WS2801 stuff. LPD8806 are very poorly documented - adafruit managed to reverse engineer enough of how they work to make an arduino library. The LPD* chips generally also do not freerun - this means your microcontroller needs a background process (timer or watchdog interrupt) pausing your program and sending out clock signals to the LEDs even when you aren't outputting colours. This clock signal runs their PWM. If you make it too fast, the colours will start to be reproduced less accurately - if you make it too slow, they will look flickery and might be an epilepsy risk.
For those reasons, I recommend you do not use LPD8806. WS2801 is a similar SPI device with a seperate chip. You can hook it up to a raspberry pi, a serial or parallel port on a computer, a digispark, littlewire, and arduino - pretty much anything capable of digitally writing to two wires. The WS2801 runs it's PWM off an internal clock, so you can set it and forget it - they'll hold their colour without needing your program to be interrupted thousands of times every second.
The WS2811/WS2812 is really neat! The WS2811 is a chip, like the ws2801, but instead of SPI it uses a purely timing-based serial protocol. They can be driven by an arduino or digispark running at 8 or 16 megahertz, they hold their colour (set and forget) just like the ws2801, and are basically the same, but you only need a single data wire instead of two, making them very appealing for use with digispark.
The WS2812 is the WS2811 chip die embedded in to a 5050-style LED, so you have an all in one addressable chainable LED module. You can buy these on 60-led per meter strips on aliexpress or from adafruit. All ws2812 LED modules run in the 800khz protocol, not the 400khz protocol available on some ws2811 devices. This means they run at twice the speed, allowing your program to communciate with them faster, so each time you send an update to your LEDs your program will spend less time doing it, and get back to your colour calculation code quicker - this potentially means a higher framerate. We also have out of the box support for 800khz ws2811 and ws2812 lights built in to the beta version of LittleWire 1.2 and included in the littlewire ruby library and c library. This means you can use a digispark to control ws2811 and ws2812 LEDs from a computer via a digispark without needing to write any arduino code or figure out how to interface with the digispark from your program - you just pass an array of colours to the littlewire libraries and it handles it all for you.
Another key difference is because of the strict timing requirements of ws2811/ws2812k, you need to precompute the colours for all of your LEDs and then use a function which outputs them from an array. You can't do streaming output, where you compute a single colour, send it out, compute the next one, etc... You can stream output with ws2801 and to some extent lpd8806 devices. This can be relevent to digispark and arduino projects because digispark only has 512 bytes of ram, and each LED takes 3-4 bytes of that memory depending on the library you use. This means you are practically limited to buffering about 100 WS2811/WS2812 LEDs in a digispark's memory. On the newer arduinos you can do a few hundred more. The Teensy 3.0 can do a bunch more still.
This limitation isn't as bad as you might think, because you can have multiple parallel strips! You fill your buffer with data for one strip and output it, then fill it with colours for a different group of LEDs and output it to a different pin! With this technique you can have roughly (100 * number of free pins) - if you have nothing else connected to a digispark it can control 600 LEDs!
In summary, I would recommend using ws2812/ws2811 (800khz) for all projects except where you want to drive LEDs from an SPI port, like that of the raspberry pi. For raspberry pi driving LEDs directly, use ws2801. Only use LPD* lights when you have to for compatibility with something else.
