Mozilla has always focused on privacy and user rights, though usually in less self-serving ways than Microsoft or Apple. Which is perhaps why Firefox is getting third-party cookie blocking now, even as Safari has had the feature for years.
A patch to block cookies coming from third-party domains has been pushed to the Firefox source code repositories.
If everything goes well, the feature should be part of Firefox 22, which has just landed in the Nightly channel so it should be arriving in the stable channel 18 weeks from now.
Most browsers, with the exception of Safari, allow websites to set cookies, pieces of data that are stored by the browser after leaving the site, but they also allow cookies coming from third-party domains, set by scripts running on the page visited.
These third-party domain cookies, more often than not, come from ad networks and they're used for tracking users across sites.
Specifically, Firefox will allow cookies from first-party origins, but will block third-party cookies unless a cookie from that domain has already been set.
This should ensure that some third-party cookies will be allowed, mostly those not belonging to advertisers, since users wouldn't normally visit the domains used to serve ads through other websites.
The policy is fairly similar to Safari's, though a bit more permissive. Mozilla hopes the change will go through without any problems, some sites may need to change a bit of code to work in Firefox, but since Safari has done the same for a decade, webmasters should be familiar with the situation.
This is just a first step, the plan is to expand the policy to other forms of content storage, for example HTML5 web storage. There are also plans to create a method for websites to request storage permissions.
Sites that respect Do Not Track may also get a more relaxed policy, since they already agree to respect Do Not Track, they may be allowed to set third-party cookies, at least some of them.