Yep, that's the ADONIS output.

You can read more about the Permutational Multivariate Analysis of Variance Using Distance Matrices.

R2 is your 'R squared' value and is just like an R2 value you would get from a linear regression, in that it explains what fraction of variance is captured by that predictor variable. (think effect size)

Pr(>F) is your p-value. As is tradition, you get to pick an alpha threshold, with both a=0.05 for 95% and a=0.01 for 99% being common.

ADONIS partitions your distance matrix and performs tests *in order*

I've written about that here: interpretation beta diversity / permdisp / adonis - #4 by colinbrislawn