I think I figured it out. I'll leave this thread up for search engine bait. I'm still curious about the UIAnchor "attach one object to the side of another" feature, though.
In NGUI, alignment is set using the pivot option that appears in the inspector. So, if you want label A to sit next to label B, and label B will be changing in size (say it's the player score), create the two labels, set the pivot of label B to "left" (click on the left arrow in the inspector just above "Depth") and then place the labels where you want them to be. You should be able to modify the text in label B and see that it remains next to label A.