In the context of e-mail sign-offs, "Regards" is a very "cold" way to end an e-mail. Other ways of ending e-mails include, "Best wishes", "Best regards", "Warm wishes", "Warm regards", "Kind regards", "Thanks", "Many thanks", etc. There are variations on these, e.g. "With best wishes" or "Yours with best wishes". I live in the UK and people have actually become quite sensitive to the tenor of e-mail sign-offs, so if someone receives an e-mail with just "Regards" at the end they sometimes even feel a little worried that the person does not like them! People will often write "kind regards" instead of just "regards" in my experience. "Cheers" is an extremely casual way to sign off an e-mail.

You don't really need to use either of them. 'Cheers' is what we say when we lift our glasses before drinking. It is also an informal way of saying 'Thanks', particularly among men in the UK. 'Regards' is a simple and rather 'cold' way of finishing a note or email before you sign your name. It's not as formal as 'Yours sincerely' and not as warm as 'Best wishes', for example. I've noticed that Spanish-speaking members here on italki often write one of these words at the end of their posts, which seems rather strange to English speakers. There is no need to use either word, and 'Cheers' seems especially inappropriate.

Regards and Best Regards are two common ways to end an email in the U.S. I use these two for strangers for the first time or two that we speak. Sincerely sounds too formal and old-fashioned for an email here in the states.

"Cheers" is a very informal way to say "thank you" (as well as a toast when having a drink with friends!) whereas "regards" is often used at the end of an email before your name. Other people might use them in other situations but that is where I come across them every day!