I want to know what $|u|=2|v|$ means in $$\{(u\#,v\#): u, v \in \{a,b\}^*, |u|=2|v|\}$$ I'm not familiar with this notation.
I also want to understand what these mean: $$\{(a^mb^n\#,a^kb^n\#): n,m,k \ge 0\}$$ $$\{(a^nb^m\#,a^kb^n\#): n,m,k \ge 0\}$$
The # signs are used to denote a end of string in Automata, aka a single # at the end of the string.
It's kinda hard to understand this question without knowing the specific context. But I'm guessing $u$ and $v$ are words in the alphabet $\{a,b\}$, and then $|u|$ and $|v|$ stand for their respective lengths. So $|u|=2|v|$ means that the length of the word $u$ is twice the length of the word $v$.
Also, things like $a^m$ represent words where the letter $a$ is repeated $m$ times. For example, $a^5=aaaaa$. If $m$ runs over all non-negative integers, then $a^m$ runs over all words (including the empty one, when $m=0$) consisting of the letter $a$ only. You can similarly interpret expressions such as $a^mb^n$.
(Disclaimer: I don't know what the "$\#$" stands for…)