I think you have the alias in the wrong spot.  Try this:

SELECT Turn1.sPrecinctID, Turn1.Turnout, (Select Count (*) from 2005Turnout
AS Turn1
Where [2005Turnout].[Turnout] > [Turn1].[Turnout];) AS Ranking
FROM 2005Turnout  ORDER BY 2005Turnout.Turnout DESC;

Dear D:

The subquery you used in the SELECT clause is almost certainly intended to
reference a separate "instance" of the table Turnout.  In order to reference
two instances of this table, you must do two things:

- Make each reference to a column in one of the two tables "explicit" as to
which table it references.

- Alias at least one of the two instances of the table, so there can BE a
distinct way of explicitly referencing each.

SELECT Turn1.sPrecinctID, Turn1.Turnout, (Select Count (*) from 2005Turnout
 Where 2005Turnout.[Turnout] > [Turn1].[Turnout]) AS Ranking
 FROM 2005Turnout AS Turn1
 ORDER BY Turn1.Turnout DESC;

In the above, I removed the ; from the subquery.

Now, outside the subqeury, it is unnecessary to make the table reference.
The additional copy of the table introduced in the subquery is not available
outside the subquery.  I might have written the whole thing like this:

SELECT sPrecinctID, Turnout,
   (SELECT Count(*) + 1
     FROM 2005Turnout T1
     Where T1.[Turnout] > T.[Turnout])
   AS Ranking
 FROM 2005Turnout AS T
 ORDER BY Turnout DESC;

Above, I've used a minimal amount of aliasing, only where necessary inside
the subquery.  I added 1 to the Ranking, as you may wish to have it begin
counting from 1 instead of 0.  That is easily removed.

Please let me know if this helped, and if I can be of any other assistance.

Tom Ellison