@reevesii's answer is great.
Another option, one that I use a lot is to use the "Draw Polygon" Tool in the Editor Toolbar to modify the Polygon you want to Fit to the boundary.
My work flow is as follows:
We have 2 polygons,Poly1 (Blue) and Poly2 (Purple). Poly1 needs to follow the boundary of Poly2. But Poly1 has too many vertices to manually edit.
Start an edit session and Select "Poly1" from the editing Templates. Using the "Construction Tools" select "Polygon" and digitise a rough polygon encompassing both Poly1 and Poly2. See below:
See how Poly1 now covers Poly2 completely? Poly1 is now two separate Polygons. We need to make them 1. Unselect all features, and then Select the two features within Poly1. If you are unsure that you have been able to select both features because your geometry is too complex, select them from the attribute table.
Once you have selected the features, head to the "Editor Toolbar", Select "Editor" and then the "Merge" tool.
It will then ask you which feature you want to combine into. In other words which attributes you want to preserve. Whichever one you click will flash on screen. So if you want to preserve the original polygon, select that one in this list.
Once you have merged the two polygons, Poly1 will be big and will cover the whole of Poly2. You now need to cut the boundary of Poly2 from Poly1. To do this simply select Poly2 on screen and again go to "Editor Toolbar" and select "Clip". If it is greyed out, make sure you have nothing else selected 1st. Also just to be safe, make sure all other layers are turned off. You want to discard all areas that overlap. See below.
Your final output will be Poly1 edited to match the outline of Poly2. This can be done with features in the same shapefile or two separate shapefiles.