Wikipedia defines the partition function as:
$$Z(\beta) = \sum_{x_i} \exp \left(-\beta H(x_1,x_2,\dots) \right)$$
Where $x_i$ are the values of random variables $X_i$, the function $H$ is understood to be a real-valued function on the space of states $\{X_1,X_2,\cdots\}$, while $\beta$ is a real-valued free parameter.
I don't understand what it is trying to say, since $x_i$ does not appear in the summand. Could someone clarify, please?
Recall that the partition function is the normalisation constant in the probability of any given state $P(x_1,x_2,...|\beta, H)$ so the sum has to be over all possible states of the system.
So, if your Hamiltonian is $H=H(x_1,x_2)$ and $X_i \in \{-1,+1\}$ for $i \in \{1,2\}$, then you have exactly four possible different states and the sum is over all of them.
There's a very nice introductory exposition in MacKay's (free) book, chapter 31.