Specifically, what is a mixed effects model and when would you use? Also, what is a test of differences in least square means and when would you use it? I searched Google but had no luck.
A mixed effects model is when you have both fixed and random effects.
I'm not 100% sure about the least-squares difference, but I believe you use it to calculate residuals. So a test would maybe find how far the residuals were from the line, so maybe to test correlation? That is my guess. If I were at home, I would look into one of my stat binders (I keep them since I minoring in stat), but I'm working at a camp for 6 weeks. Try wikipedia.