Two aggregated sub queries wont run
Try this, it should help put you in the right direction.

SELECT l.StaffID, 
       l.FirstName, 
       l.Surname, 
       m.ModuleDescription
 FROM dbo.Lecturer as l
 RIGHT OUTER JOIN dbo.Module as m
   ON l.StaffID = m.ModuleConvenor
WHERE m.ExamWeight = (
            SELECT AVG(m2.ExamWeight) AS 'Avg ExWeight'
              FROM dbo.Module as m2
             WHERE m2.ModuleConcenor = l.StaffID OR m2.ModuleConvenor IS NULL) 
 AND m.CWWeight =  (
    SELECT AVG(m3.CWWeight) AS 'Avg CWWeight'
     FROM dbo.Module as m3
    WHERE m3.ModuleConcenor = l.StaffID OR m3.ModuleConvenor IS NULL)


I did the following


Updated your where clause syntax
Update your aggregate function typo from AVE to AVG
Updated your alias being used in the aggregate function in both subqueries


You may need to post some sample data and table ddl information. It is not clear to me how the module and lecturer tables should be joined based on your query. This answer helps correct your syntax, it may not be full solution to what you are trying to achieve.