Date_Column is just that. Whatever column in your table that contains data of the datetime data type (or any other data type that can store date data). Given your example using your calendar table this would be the column "date". (Note: In some RDBMS's date is a reserved word so you may want to consider naming such columns in the future something different like EntryDate)
So replacing date_column with date would result in:
$query = "SELECT DATE_FORMAT([b]date[/b], '%m %d %Y') as FormatedDate FROM calendar";
You may also want to add a seperator in your format string so instead of '%m %d %Y' which will return 01 12 2003 you may want to add in a "-" to make the string look like '%m-%d-%Y' and return 01-12-2003.
To display this data using php
$query = "SELECT DATE_FORMAT([b]date[/b], '%m-%d-%Y') as FormatedDate FROM calendar";
$result = mysql_query($query);
$entry = mysql_fetch_array($result);
echo = $entry['FormatedDate'];
or if you know your only getting back 1 row of data
$query = "SELECT DATE_FORMAT([b]date[/b], '%m-%d-%Y') as FormatedDate FROM calendar";
$result = mysql_query($query);
echo mysql_result($result, 0, "FormatedDate");