Why does $L = \{ 0^n 1^n \space | \space n \in \mathbb{N} \}$ belong to $\mathrm{P}$?
You can easily write a polynomial-time algorithm to decide&nbsp;$L_1$ (just count the zeros and check you see the same number of ones), so the language is in $\mathrm{P}$.

Most non-regular languages are not in&nbsp;$\mathrm{P}$ because there are uncountably many languages in total, but only countably many languages in&nbsp;$\mathrm{P}$.

The halting problem is not in&nbsp;$P$. If you want a computable example then, by the time hierarchy theorem, any language that is $\mathrm{EXP}$-complete is not in&nbsp;$\mathrm{P}$.

This answer is intentionally brief because all the things I've mentioned are well-known facts that you should be able to look up, now that you know what to look for. If you need more detail, I suggest asking a new question about that, after checking that we don't already have a question about it.