Transforms change the geometry and/or the shape of a layer. The most basic transforms are scale or flip which simply change the size or the orientation of a layer. Mapping and warping transforms re-shape the layer in such way that you can composit the transformed layer on top of another layer in order to match it. This way you can achieve the effect that the trnasformed layer "belongs" to the base layer.
Transformtions that map or warp a layer can be motion animated either by supplying a motion data source or by animating the parameters of the transform. Motion data can be exported from the FX Editor or from Mocha, or AfterEffects.