Emails and email are both correct plurals, but each has its own context. It depends on whether or not you are using it as a countable or uncountable noun.
You can use email as an uncountable noun, just like mail. For example, "I received lots of email today" or "John sends me too much stupid chain email". But, you cannot use email as a countable noun. For example, "I have five email" makes as much sense as "I have five mail".
You can use emails as a countable noun, just like letters. Originally, email was always uncountable, just like mail. But over time, largely due to the ubiquity of use, pieces of email or email messages (and similar phrases) were shortened, and simplified to just emails.
For example: "I have five emails" or "Kevin sends me too many emails". (Contrast that last one with "Kevin sends me too much email".)
Of course, you can still use emails in some contexts where email is also acceptable ("I received lots of emails today"), but it does have a slightly different meaning (compare to "I received lots of email messages today") because you are using a countable versus uncountable noun.