What is the best free third-party email client for Mac?

Other than Apple's Mail client, there is Thunderbird by Mozilla.

As of 2016, it seems that Mac users are limited. Many of the third-party email clients that are claimed to be the best by web reviews come at a cost of $10-20. The clients typically listed in this space are Airmail, Postbox, Mailmate, Cloudmagic, Unibox, Outlook, among others. Airmail is frequently cited as the best, and from there opinions tend to varies. Now, the best free email clients are probably https://www.mozilla.org/en-US/thunderbird/, which has an interface similar to Firefox. Personally, I'm not a fan of the Mozilla interface style, rather I prefer the apps that share the new sleeker Mac style. Nonetheless, functionally speaking, Thunderbird has most everything I can think of. It has good support for various email hosting services, has to-do list and calendar support, a unified mailbox, and many other features (which I don't use at the moment, so I can't really talk about). Also, there is https://www.nylas.com/n1, an open-source email client for OS X, Windows, and Linux which is relatively new. It doesn't have all the functions that other clients have (yet), but it will likely come as the user base grows and more contributors get involved. It has a sleek interface and all the basic functionality, but doesn't yet support things like an unified mailbox, or calendar support.At this point, there are slim pickings for free, quality, third-party clients for Mac. This is largely due to large tech companies buying up email companies and shutting them down (Google bought Sparrow, a very popular client, and shut it down).  Hopefully, this space will grow again in the future.

There is also Sparrow Mail: http://sparrowmailapp.com/ I'm not fond of the interface and from what I remember, it only works with google accounts. But it is a free alternative. Note, Opera (the web browser) also has email functionality: http://www.opera.com/mail/

There is a free version of the newzapp software called Flex  http://www.newzapp.co.uk/newzapp-packages-flex.asp And although it's currently Windows based, once you have signed up just contact them and ask them for access to newzapp 5 (the new cross-browser, MAC friendly version which is out now in beta mode).