I'm not completely sure why the "char" parameter isn't working. The table field "LastName" is nvarchar(50). A "char" can be implicitly converted to an "nvarchar".
But converting your "char" parameter to an "nvarchar" first will solve your problem:
CREATE PROCEDURE FindEmployee @EmpLName char(20)
AS
DECLARE @LName nvarchar(20)
SET @LName = RTRIM(@EmpLName) + '%'
--SELECT @EmpLName = RTRIM(@EmpLName) + '%';
SELECT c.FirstName, c.LastName, a.City
FROM Person.Contact c JOIN Person.Address a ON c.ContactID = a.AddressID
WHERE c.LastName LIKE @LName --@EmpLName;
GO
EXEC FindEmployee @EmpLName = 'Barb';
GO