Let’s start with the “why should I care” part. Many email servers on the Internet will check the sending server of all email received, and if that server does not have “reverse DNS” set up properly, the email will be refused! So you could be sending legitimate email to people who have asked to receive it, and the email could still bounce just because of some server/domain settings that you never even knew about!
So, what is reverse DNS then? Think of it like this. To get to a specific website, you normally type in the domain name. The domain name must be resolved to a unique address in order for the website to be found by your browser. Domain name servers, which are registered when you set up a domain name for your website, are servers that know which address your website is at.
Reverse DNS is just the opposite. It takes a website address, and figures out what domain that address belongs to. In order to do this, the domain name for the address has to be registered by the Internet Service Provider (ISP) managing that address. Many times, the ISP doesn’t register this until you ask them to. After all, they don’t know what domain name will be associated with an address until you order it. But also in many cases, shared server owners don’t bother (or forget) to register the reverse DNS.
In additon, if the reverse DNS isn't set up properly, an error can occur during the lookup. If an error occurs, regardless of the error detail, many email servers will bounce the email. An example of a common error is the following: Your ISP has set up reverse DNS for the IP xxx.xx.xxx to point to example.com. But example.com points to the IP yyy.yy.yyy. Despite the reverse DNS being set up, the complete check results in an error. The IP must point to a domain, and that domain must point back to the same IP. If it doesn't, reverse DNS lookup will fail, and this will result in many bounced emails.
So, it might be a good idea to take a moment and check your IP to make sure the DNS is registered. You can perform this check at Domain Tools. Just enter your IP address, and see if it pops up your domain name (or if your site is on a shared server, a bunch of domain names).
If reverse DNS is not set up, contact your ISP now to get it set up!