The first thing to establish is what exactly you need to do, to decide whether or not a framework and/or MVC (or other design pattern) would be of benefit.
Frameworks are there to provide a consistent platform for development whilst usually providing solutions to common programming requirements (such as Database interaction, form creation and validation, user authentication etc.)
For PHP at least the MVC / HMVC design pattern does tend to dominate the mainstream frameworks available (e.g. Zend, CakePHP, CodeIgniter etc.) but there are many different design patterns that one could use.
MVC is so popular because it offers an established and understood way of separating data modelling and processing logic from view/presentation layer (something which is considered desirable in order to produce robust, scalable applications).
It's important to note (and as was expressed by @Marjan Venema in a comment to @ElYusubov's answer) that MVC, MVP, MVVM and the other MVx patterns are (in principal at least) all the same 'design pattern'.
Typically different design patterns all serve (often subtly) different purposes and in several cases were developed with a specific language in mind. However a true 'design pattern' is not a hard and fast rule to programming and is really more of a philosophical / idealogical understanding of a programs implementation and design requirements and logical function(s).
Research is the best way to find out about different programming principals and best practices, here's some Wikipedia links to get you started:
In practice there's nothing stopping you from implementing your own 'pattern', IMO the best way is to learn by doing, for me at least I didn't fully understand the MVC pattern until I started trying to write a web site using it.
Once you understand some of the programming concepts and best practices you can use those to build your own system to solve the specific problems you're facing and to meet your needs, whether it conforms to an established 'pattern' or not.
If you have no specific set of problems to solve then learning one of the common frameworks is your best bet.