TagBox[boxes, tag] displays as boxes but maintains tag to guide the interpretation of boxes on input.
TagBox provides a way to store hidden information in Mathematica output.
TagBox is generated sometimes in StandardForm output, and often in TraditionalForm output.
By convention, tag is typically a symbol that corresponds to the head of the interpreted form of boxes.
The following options can be given:
If you modify the displayed form of TagBox[boxes, tag] only boxes will be modified, and there is no guarantee that correct correspondence with expr will be maintained.
See The Mathematica Book: Section 2.8.10.
See also: InterpretationBox, FormBox, ToExpression.