ARMA & MA methods: how do you know the error terms? Reading the ARMA model for the first time, and I'm confused.

Let's say I have a time series

x = [1, 2.1, 2.9, 3, 4.1]


According to the ARMA model, $X_t$ is a linear combination of previous values and errors, something like

$$\sum_i \phi_i X_{t-i} + \sum_i \theta_i \epsilon_{t-i}$$

But, what are $X_i$ and $\epsilon_i$??


is $X_i$ the actual value of the series at $t=i$? Eg, in my example $X_2 = 2.1 ?? (with 1-based indexing)
if so, what are the error terms?


The same question for the simple moving-average model, where all the $X_i$ values are remain unused.
In the context of discrete-time statistical signal processing, an ARMA-(p,q) random process is defined as (assuming zero initial conditions)

$$ \sum_{k=0}^p a_k ~ x[n-k]  = \sum_{k=0}^q b_k ~ v[n-k] $$ 

or equivalently

$$ x[n] = - \sum_{i=1}^p a_i ~ x[n-i]  + \sum_{i=0}^q b_i ~ v[n-i] $$ 

where $v[n]$ is a white-noise (WSS) random process with variance $\sigma_v^2$ and $x[n]$ represents the resulting ARMA process.

In this context your coefficients are related as $$\phi_i = - a_i $$ and $$\theta_i = b_i$$ The process reggresses over its past values $x[n-i]$ ($X_{t-i}$ acc) and is also a moving average of the input noise $v[n-i]$ ($\epsilon_{t-i}$ acc).