Here's a short video of how 23andMe processes the DNA. In a nutshell, the saliva only holds 0.5% DNA, so they amplify it, embed it into a special chip, and analyse it.
Here's my ancestry breakdown at the highest certainty level of 90%:
Things get more interesting when the strictness is reduce. Here's my ancestry breakdown at the lowest certainty level of greater than 51%:
Here's some information on traits:
And here's the neanderthal DNA report. Finally, I'm above average in something!
Note, this is just small sampling of the information provided.