By wrapping with single quotes, you've created the problem. You're instructing MySQL to try to order by the word "answered" (rather than the column in the database). Simply remove the quotes.
SELECT * FROM contact ORDER BY answered ASC
As pointed out in the comments, you may also surround the column names with `backticks` like
SELECT * FROM `contact` ORDER BY `answered` ASC
Backticks are to be used for table and column identifiers, but are only necessary when the identifier is a MySQL reserved keyword, or when the identifier contains whitespace characters or characters beyond a limited set (see below) It is often recommended to avoid using reserved keywords as column or table identifiers when possible, avoiding the quoting issue.
When to use single quotes, double quotes, and backticks in MySQL