I'm using GNUS to read my emails. Is there some way to make it automatically check my mail box and report new mails somehow? Maybe some package exists to do this sort of thing?
You'll want to look at Gnus Demon.
gnus-demon.el you'll probably want
gnus-demon-add-scanmail [Defaults to 120, 60: Scan every 2 hours if idle for 1 hour].
This is probably closer to the standard 5-minute scan on phones/applications:
(gnus-demon-add-handler 'gnus-demon-scan-mail 5 t)
EmacsWiki has a comparison of Gnus Notifications, some are for modeline alerts to new email, some are for desktop notifications.