The normal (Gaussian) p.d.f is given by $$f(t)=\frac{1}{\sqrt{2\pi}\sigma}\exp\big(-\frac{(t-\mu)^2}{2\sigma^2}\big)$$ and a random variable $T$ with this distribution is denoted by $T\sim n(\mu, \sigma^2)$. If $Y_1,\ldots,Y_n \sim n(\mu,\sigma^2)$ are independent then the joint density is the product of the individual p.d.f's
$$\prod_{i=1}f(y_i)=\frac{1}{(2\pi)^(n/2)\sigma^n}\exp\big(-\frac{1}{2\sigma^2}\sum_{i=1}^n(y_i-\mu)^2\big)$$
For fixed data, $y_1,\ldots,y_n$, this can be viewed as a function of the parameters denoted $L(\mu,\sigma^2)$ where $L$ is termed the likelihood function.