I understand the $f_n(x_1,…,x_n|\theta)$ and $X_1,…,X_n$ statements but what confuses me is the part about the unknown parameters $\theta = (\theta_1,…,\theta_k)^T$ .
Is there set of parameters, $\theta$, unique to each instance of $x_i$ in $X$? Or does each instance of theta correlate with a unique $x_i$ in $X?$ (If so, why is $k$ used instead of $n$ [I don't think this is the case])
Is this a general definition that applies to scalar and vector instances of $\theta$?
If this was a $gamma$ distrubution, I would assume that each instance of $\theta$ would be a vector of 2 values (i.e. $\alpha$, $\beta$) or if it was a gaussian distribution, it would be a mean and standard deviation. Is this correct?
You are right. This definition means that X is a random variable that depends on $\bf{\theta}=(\theta_1, ..., \theta_k)$. And $x_1, ..., x_n$ are n independent draws from the random variable $X$
Let me give you an example, if X is normal distributed, then $\theta = (\mu, \sigma^2)$ would be the parameter vector of $X$. You can draw as many samples from this normal distribution as you like, and these samples are all identically distributed by $N(\mu, \sigma^2)$