Update a table using JOIN in SQL Server?

Try it like this:

    UPDATE a 
    SET a.CalculatedColumn= b.[Calculated Column]
    FROM table1 a INNER JOIN table2 b ON a.commonfield = b.[common field] 
    WHERE a.BatchNO = '110'

You don't quite have SQL Server's proprietary UPDATE FROM syntax down. Also not sure why you needed to join on the CommonField and also filter on it afterward. Try this:

UPDATE t1
  SET t1.CalculatedColumn = t2.[Calculated Column]
  FROM dbo.Table1 AS t1
  INNER JOIN dbo.Table2 AS t2
  ON t1.CommonField = t2.[Common Field]
  WHERE t1.BatchNo = '110';

If you're doing something silly - like constantly trying to set the value of one column to the aggregate of another column (which violates the principle of avoiding storing redundant data), you can use a CTE (common table expression) - see here and here for more details:

;WITH t2 AS
(
  SELECT [key], CalculatedColumn = SUM(some_column)
    FROM dbo.table2
    GROUP BY [key]
)
UPDATE t1
  SET t1.CalculatedColumn = t2.CalculatedColumn
  FROM dbo.table1 AS t1
  INNER JOIN t2
  ON t1.[key] = t2.[key];

The reason this is silly, is that you're going to have to re-run this entire update every single time any row in table2 changes. A SUM is something you can always calculate at runtime and, in doing so, never have to worry that the result is stale.