transformation: A transformation is a change, generally in the context of a coordinate system. In this context, the transformation will change the basis set which defines the dimensions of the coordinate system. The two basic types of transformations are translations and rotations. An example of a rotation transformation is to switch from considering North-South versus East-West to considering Northeast-Southwest versus Northwest-SouthEast. (The rotation is usually about the origin, but it can be about any point.) An example of a translation transformation is to switch from considering Chicago as the origin to considering New York as the origin.
translate: To translate is to shift or to adjust by "sliding" in a particular direction. There are two contexts for this. First, an object which is translating is moving along a straight line in a given direction. Second, if one translates a coordinate system, then the origin of that coordinate system is shifted. As an example of the second case, imagine a map of a city as a coordinate system and if we allow the mayor to consider City Hall as the origin and the bank teller to consider the bank as the origin, then we can translate between these two pictures. Assume that the bank is two blocks east of City Hall and that a grocery store is three blocks east of City Hall. The Mayor tells you the store is "three blocks east" ("of City Hall" is implied since it is the Mayor's origin) while the bank teller tells you the store is "one block east" ("of the bank" is implied since it is the bank's origin). We translate from the Mayor's coordinate system to the teller's by shifting by two block to the east.