Oracle SQL - DATE greater than statement

As your query string is a literal, and assuming your dates are properly stored as DATE you should use date literals:

SELECT * FROM OrderArchiveWHERE OrderDate <= DATE '2015-12-31'

If you want to use TO_DATE (because, for example, your query value is not a literal), I suggest you to explicitly set the NLS_DATE_LANGUAGE parameter as you are using US abbreviated month names. That way, it won't break on some localized Oracle Installation:

SELECT * FROM OrderArchiveWHERE OrderDate <= to_date('31 Dec 2014', 'DD MON YYYY',                           'NLS_DATE_LANGUAGE = American');

You need to convert the string to date using the to_date() function

SELECT * FROM OrderArchiveWHERE OrderDate <= to_date('31-Dec-2014','DD-MON-YYYY');

OR

SELECT * FROM OrderArchiveWHERE OrderDate <= to_date('31 Dec 2014','DD MON YYYY');

OR

SELECT * FROM OrderArchiveWHERE OrderDate <= to_date('2014-12-31','yyyy-MM-dd');

This will work only if OrderDate is stored in Date format. If it is Varchar you should apply to_date() func on that column also like

 SELECT * FROM OrderArchive    WHERE to_date(OrderDate,'yyyy-Mm-dd') <= to_date('2014-12-31','yyyy-MM-dd');

you have to use the To_Date() function to convert the string to date !http://www.techonthenet.com/oracle/functions/to_date.php