Get last login time of each User ID
You need a simple MAX:

SELECT DATE_FORMAT(MAX(lastlogin), '%d-%b-%Y') AS lastlogin,
   user_id
FROM last_logins
WHERE user_id IN (100,110)
GROUP BY user_id
ORDER BY MAX(last_logins.lastlogin) DESC


Don't sort by the formatted lastlogin as this is not a date anymore.

Don't alias using an existing column name, this might lead to confusion and well-hidden bugs (the same condition in WHERE and HAVING meaning different things, for example).

See SQL Fiddle by jkavalik