to_timestamp()
You need to use to_timestamp() to convert your string to a proper timestamp value:
to_timestamp('12-01-2012 21:24:00', 'dd-mm-yyyy hh24:mi:ss')
to_date()
If your column is of type DATE (which also supports seconds), you need to use to_date()
to_date('12-01-2012 21:24:00', 'dd-mm-yyyy hh24:mi:ss')
Example
To get this into a where condition use the following:
select * 
from TableA 
where startdate >= to_timestamp('12-01-2012 21:24:00', 'dd-mm-yyyy hh24:mi:ss')
  and startdate <= to_timestamp('12-01-2012 21:25:33', 'dd-mm-yyyy hh24:mi:ss')
Note
You never need to use to_timestamp() on a column that is of type timestamp.