Easing in means when you are trying to piece together two units that don't match exactly.
For example, when we attach our final borders to a quilt it is best practice to measure the center of your quilt top for the border measurement then cut your border piece to match that measurement. During construction process things stretch and go off fractions of inches. So your edge may measure 50 3/4" while the center of your top and your cut to that measurement border piece may only measure 50 1/2". You need to "ease in" the larger measurement piece. I am old school so I do it by pinning in the center, pinning each corner, then pin at the half way point between center and corner and continue doing that until the whole unit is pinned about every 3 to 4" or so then I end up distributing that extra 1/4" across the entire length of the unit instead of in just a small area resulting in a pleat or gather where you don't want one.
Easing in is also used when doing curved piecing. It is done a lot in clothing construction. here is a link to a you tube video
http://www.youtube.com/watch?v=Eh7_zvHcIio
PS love your dog in your avatar. What a beauty!