I have written a Python script that checks a certain e-mail address and passes new e-mails to an external program. How can I get this script to execute 24/7, such as turning it into daemon or service in Linux. Would I also need a loop that never ends in the program, or can it be done by just having the code re executed multiple times?
You have two options here.
Make a proper cron job that calls your script. Cron is a common name for a GNU/Linux daemon that periodically launches scripts according to a schedule you set. You add your script into a crontab or place a symlink to it into a special directory and the daemon handles the job of launching it in the background. You can read more at Wikipedia. There is a variety of different cron daemons, but your GNU/Linux system should have it already installed.
Use some kind of python approach (a library, for example) for your script to be able to daemonize itself. Yes, it will require a simple event loop (where your events are timer triggering, possibly, provided by sleep function).
I wouldn't recommend you to choose 2., because you would be, in fact, repeating cron functionality. The Linux system paradigm is to let multiple simple tools interact and solve your problems. Unless there are additional reasons why you should make a daemon (in addition to trigger periodically), choose the other approach.
Also, if you use daemonize with a loop and a crash happens, no one will check the mail after that (as pointed out by Ivan Nevostruev in comments to this answer). While if the script is added as a cron job, it will just trigger again.