Learn Turkish: Örneğin bu kural çok önemli. - For example this rule is very important.

In Turkish, the present tense "is" or "are" often doesn't need a verb or suffix for the third person (he, she, it, or this). While you could add the suffix -dir to the end to make önemlidir, in everyday language we usually just drop it. Saying bu kural çok önemli is a complete, grammatically correct sentence.

Is there a difference between örneğin and mesela?

They both mean "for example" and are used interchangeably. Mesela is an older Arabic loanword that is extremely common in everyday speech. Örneğin is a native Turkish word built from the root örnek (example). It is great to know both because you will hear both all the time!

Could I just say örnek bu kural instead of örneğin?

No, you cannot. The word örnek is a noun meaning "example". To use it as the introductory phrase "for example", you must use the special adverb form örneğin. Another correct option would be to say örnek olarak, which translates to "as an example".

If the noun was plural, would bu change to bunlar?