An explanation is describing a complex idea so someone can understand it.
An elaboration is providing more information to construct a complete description.
"The teacher's explanation of the problem was pretty good; she clarified the problem statement, and showed how the problem could be solved. At that point, though, only some of the students understood. She then gave an elaboration, describing how similar problems could be solved, and showed more examples. With the more elaborate answer, all the class understood."
There's only one "i" in explanation.