want date format as mm/dd/yyyy – 10/02/2008

The SQL CONVERT function converts between formats - and the DATETIME specifier 103 does indeed generate "mm/dd/yyyy".
Unfortunately, it only works when you feed it a DATETIME value to convert to a VARCHAR or NVARCHAR value - so if you are getting "Dec 29 2015 12:00AM" from your SELECT, then that's because you are storing your dates in VARCHAR or NVARCHAR fields already, and SQL won't convert from a string based field to a string based field.
Try it yourself:

SELECT convert(nvarchar, GETDATE(), 103) as 'Date Of Birth1',
       convert(nvarchar, GETDATE())as 'Date of Birth2',
       convert(nvarchar, convert(nvarchar, GETDATE()), 103) as 'Date Of Birth3'

Always store values as the appropriate datatype: anything else will give you problems later.
Change your database design now, before it does become a problem to fix - or at some point an invalid date value will get in there, and it will get to be a real problem to sort out!