If $X_1, X_2, X_3, \cdots$ is a sequence of independent identically distributed random variables with $E[X_i] < \infty$ and $Var(X_i)< \infty$ such that the sequence $Y_n = 3 \frac{X_1 + X_2 + \cdots + X_n }{\sqrt{n}}$ converges in law to a standard Normal distribution, compute $E[X_i]$ and $var(X_i)$.
As it is used in this question, what does it mean for this sequence to converge in law to a standard normal? I know law is synonymous with distribution but I still don't fully understand the implications of a convergence in law.
It means that the CDF of $Y_n$ converges pointwise to the CDF of a standard normal at all points of continuity of a standard normal CDF (which is on all of $\mathbb{R}$).
Convergence in law is also sometimes called convergence in distribution or weak convergence.
In this case, note that if $X_1,\ldots,X_n$ are i.i.d. with mean $\mu$ and variance $\sigma^2<\infty$, then defining $S_n = \frac{1}{n} \sum_{i=1}^n X_i$, we have $\frac{\sqrt{n} (S_n - \mu)}{\sigma} \implies N(0,1)$ where $\implies$ denotes convergence in distribution (this is a Central Limit Theorem).
You're supposed to match the form of $Y_n$ to $\frac{\sqrt{n} (S_n - \mu)}{\sigma} $ and read off $\mu$ and $\sigma$ to solve this problem.