Firstly, format your dates to the ISO-8601 standard. Wrap it in Date() to ensure it gets processed as a DATE. Finally, construct your range so that it will include everything from 12:00am onwards until just before 12:00am the next day.
select itemId, dateColumn
from items
where dateColumn >= date('2012-10-23')
AND dateColumn < date('2012-10-23', '+1 day')
SQLite columns are not typed. However, if you compare the column to a DATE as shown, it is sufficient to coerced the column data into dates (null if not coercible) and the comparison will work properly.
Example on SQLFiddle:
create table items (
itemid, datecolumn);
insert into items select
1,'abc' union all select
2,null union all select
3,'10/23/2012 12:23' union all select
4,'10/23/2012' union all select
5,'2012-10-23 12:23' union all select
6,'2012-10-23' union all select
7,'2012-10-24 12:23' union all select
8,'2012-10-24' union all select
9,date('2012-10-24 12:23') union all select
10,date('2012-10-24');
Results:
itemid datecolumn
5 2012-10-23 12:23
6 2012-10-23
Note that although rows 3 and 4 appear to be dates, they are not, because they do not conform to ISO-8601 formatting which is the only format recognized by SQLite.
WHERE-clause to select only dates matching X