[Update] According to reports via iMore.com it now appears as though the new beta program may not be a true public beta test, instead it may be invite only. Editors at iMore refer to an email they received that has also been confirmed to be sent out to a handful of people who were already signed up to the early-access beta program. The email was a direct invitation to them to participate in the iOS 8.3 beta. As mentioned below there were some issues with some users accessing the beta. So this could explain it as not everyone that has attempted to register has received this invitation - and of course, if the beta was available to everyone, it would make very little sense to send such an email out to only a handful of people.
For those looking to checkout the latest improvements and changes to iOS, Apple has launched a new public beta testing program that will allow anyone that is willing to test out the latest versions of the company's mobile software.
What is the Apple Beta Software Program?
The Apple Beta Software Program lets users to try out pre-release versions of software, applications, and services. The feedback you provide on quality and usability helps us fix issues and make Apple software even better. Please note that since the public beta software has not yet been commercially released by Apple, it may contain errors or inaccuracies and may not function as well as commercially released software. Be sure to back up your Mac using Time Machine and your iOS device with iTunes before installing beta software. Install only on non-production devices that are not business critical. We strongly recommend installing on a secondary system or device, or on a secondary partition on your Mac.
You can get started in the program by signing up on the Apple Beta Software Program site. Before downloading any of the new software Apple asks that users backup their device to iTunes and then download a configuration file which will let the beta show up on the Software Update screen.
Just note that the only way to downgrade to the current shipping version of iOS (8.2, at this writing) is to completely wipe your device and restore that backup. Otherwise your phone or tablet will continue to install beta builds and, eventually, the final release of iOS 8.3.
According to 9to5Mac the program is still in the roll out phases and not everyone has been able to access all parts of the site. However, they have put together a great overview of the sign-up and installation process. Once you've signed up with your Apple ID, you have to install a profile from appleseed.apple.com/profile, which points your iPhone or iPad at the beta software update channel instead of the standard channel. You then download and install the newest iOS 8.3 beta as you would any over-the-air update.
For more details on the program head over to the FAQ. It should be noted that beta software, even that from Apple, can often times be very buggy and is generally not completely stable. There is also going to be limited support if there are any issues.
Apple has traditionally limited beta software access to its registered developers - anyone can sign up for an account whether they’re actually intending to develop apps or not, but the OS X and iOS programs each cost $99 a year.