Learn Russian: Родной язык — русский.

In Russian, the present tense of the verb 'to be' (am, is, are) is almost always left out. When a sentence simply states that one thing is another, writers often use a dash to show where the word 'is' would go. Here, it connects Родной язык (native language) to русский (Russian).

Why do родной and русский have different endings if they are both describing a masculine word?

They are both masculine adjectives matching the masculine noun язык, but they belong to different spelling groups. The ending -ой in родной is used because the stress falls on the last syllable. The ending -ий in русский is a standard unstressed masculine ending. Both do the exact same grammatical job here.

Does русский mean the Russian language or a Russian person here?

Here it means the language. The word русский is an adjective that literally means 'Russian'. When talking about languages, you can say русский язык or just русский as a shorthand, and people will know you mean the language from the context of the sentence.

Does the word язык mean anything else?

Yes! Just like the older use of the English word 'tongue' (as in 'mother tongue'), the Russian word язык means both a spoken language and the actual physical body part in your mouth.