When editing a mesh, I can select a vertex and move it, and it moves independently of the other vertices. However, I'd like for it to move nearby vertices with it, and the pull drop off with distance. For example, if I could select the vertices at the corner of the mouth and move them around, slightly modifying the upper lip, chin, and cheek slightly.
Use proportional editing.
Proportional Edit is a way of transforming selected elements (such as vertices) while having that transformation affect other nearby elements. For example, having the movement of a single vertex cause the movement of unselected vertices within a given range. Unselected vertices that are closer to the selected vertex will move more than those farther from it (i.e. they will move proportionally relative to the location of the selected element). Since Proportional Editing affects the nearby geometry, it is very useful when you need to smoothly deform the surface of a dense mesh.