Hugo Kornelis continues a quest to document query plan operators:
The Compute Scalar operator is used to compute new values based on other columns in the same row. These new values are then added as extra column(s) in the output rows.
The expressions used to compute the new values can only refer to constant values and to columns in the input rows of the Compute Scalar operator. Other than that, there are, to my knowledge, no restrictions. The expressions can vary from very simple to extremely complex. The expressions can even include references to scalar user-defined functions, to CLR user-defined functions, and to built-in CLR functions.
Read on for a good deal of information about the operator.