Look at what are called shape keys or, previously, vertex keys.
There are two basic types: “absolute” and “relative.” Both are well-documented in the Blender manual (at the Blender site) and elsewhere, so I’ll just give you a high-level overview of the concepts.
Absolute keys are great for “morphing” a mesh (or anything else) from one position to another. You could use these to move the sphere from its normal to its dented shape; ditto (and separately…) the thing that rammed into it. To do this, you first mark a “basis key,” which is the un-deformed shape, then you define one or more subsequent keys which are variously deformed shapes. Finally, an IPO named “speed” is used to control the morphing from one shape to the next. At any point in time, the IPO is positioned somewhere between one-or-two keys, and the shape is therefore somewhere in the process of shifting from one to the other as determined by the IPO.
Relative keys (you may encounter the term, “RVK”) are great for animating things like facial-expressions, since individual groups of vertices can be easily controlled and more-than-one key can be applied in arbitrary amounts at the same time. Hence, we might make Suzanne wink while wiggling her ears and smiling, before shifting to an astonished look with no wiggle. The process is once again based upon an undeformed “basis key.” In this case, each key has an IPO-curve which determines the strength (positive or negative) with which this key is applied to the shape.
(As you undoubtedly know, “an IPO” is a curve, usually a Bezier curve, which defines the change of some value [the Y-axis] over time [the X-axis]. IPOs are used throughout Blender.)