Boot cut means a slightly flared leg (think a very very subtle bell bottom) so that they fit over cowboy or work boots easier.
Relaxed fit means it's made a little bigger in the seat and thighs compared to the same size in a regular fit. The lines/silhouette aren't as clean/nice, but it's comfier; which is why it's called relaxed fit.
Soft face is exactly what it says, it's got soft rubber faces, basically just a shrunk down rubber mallet. Use it on things you don't want to leave marks on.
Double peen is basically just a soft face except one side is like a hard epoxy like material. Harder than the rubber side, still softer than a metal head would be.