Swiftmailer and PHPMailer are probably the two most popular 3rd party mail packages for sending email from within PHP code. I use PHPMailer mainly because it's the one I know, but I know more than a couple people who prefer Swiftmailer -- but whichever you use will get the job done, I think.
The advantage of using them over the native mail function is that they take care of a lot of the detailed grunt work so that you don't have to. They make it much easier to do things like sending HTML email (with plain text alternative content) and file attachments, sending via a separate SMTP account on another server, taking care of setting the right headers and dealing with basic email security concerns, and so forth. It's nothing you couldn't do yourself with built-in PHP -- after all, PHPMailer is nothing but a library of PHP code -- but unless you want to do all the work yourself as some sort of learning exercise, why bother?