By using the same field in the PARTITION BY and ORDER BY clause, the rn field will always equal 1.
Assuming that new_registration_no = code and category = decs, you could change the ORDER BY field to be ORDER BY category DESC to get that result. However, that's a pretty arbitrary ORDER BY - you're just basing it on a random text value. I'm also not 100% sure how well the ROW_NUMBER() function works in a CTE.
A better solution might be something like:
SELECT *
FROM
(
SELECT
New_Registration_No,
Category,
ROW_NUMBER() OVER
(
PARTITION BY New_Registration_No
ORDER BY
CASE
WHEN Category = 'Re-Registration' THEN 1
WHEN Category = 'New' THEN 2
ELSE 3
END ASC ) rn
FROM Equipment_Registrations
) s
WHERE rn = 1
You can set the order in the CASE statement to be whatever you want - I'm afraid that without more information, that's the best solution I can offer you. If you have a known list of values that might appear in that field, it should be easy; if not, it will be a little harder to configure, but that will be based on business rules that you did not include in your original post.