Trouble understand HMM Forward Algorithm Referencing:
https://en.wikipedia.org/wiki/Forward_algorithm
I understand most of the expansion of the forward algorithm, but the very first step is confusing to me:

Why does the joint probability of current and prior emissions and current state, equal to some term of summation over prior states? What does that summation mean? What is it summing over? The t value?
 A: That first step is just an instance of the sum rule: 
\begin{align}
P(A = a) = \sum_{b} P(A = a, B = b)
\end{align}
where the summation is overall possible values of $B$. This is true for any random variables $A$ and $B$, as long as you sum over all possible values of $B$. This rule is actually pretty intuitive when stated in words. Suppose $A=a$ represents the event that it rains tomorrow, and $B$ represents whether or not humans will ever colonize Mars. Then, the sum rule states that the probability that it rains tomorrow is:


*

*The probability that it rains tomorrow and humans do eventually
colonize Mars, plus

*The probability that it rains tomorrow and humans do not ever colonize Mars.


For HMMs, the state at each time step is a distinct random variable, and so we could pick any time step to sum over. The forward algorithm chooses to sum over the state of the previous time step because the conditional independence assumptions of HMMs allow the rest of the derivation to proceed conveniently.
You'll encounter the same idea with the backward algorithm, which chooses to sum over the state of the next time step.
\begin{align}
\beta_t(x_t) = P(y_{t+1,n} | x_t) = \sum_{x_{t+1}} P(y_{t+1:n}, x_{t+1} | x_t)
\end{align}
where $n$ is the length of the string.
Both of these algorithms together form the forward-backward algorithm for computing the probability of a state at time $t$ conditional on the whole string:
\begin{align}
P(x_t | y_{1:n}) & =
\frac{
P(x_t , y_{1:n})
}{
P( y_{1:n} )
} \\
& = 
\frac{
  P( x_t, y_{1:t} ) P( y_{t+1:n} | x_t)
}{
  P( y_{1:n} )
} \\
& =
\frac{
  \alpha_t(x_t) \beta_t(x_t)
}{
  \sum_{x_n}\alpha_n(x_n)
}
\end{align}
In short, the Forward-Backward algorithm uses the conditional independence assumptions of HMMs to compute marginals efficiently by picking an order for summing out random variables. It is a special case of the Sum-Product algorithm, which does the same thing for graphical models more generally.
A: For simplicity say that all the random variables are discrete. Then 
\begin{align*}
P(X_t = x_t , Y_{1:t} = y_{1:t}) &= P(X_t = x_t , X_{t-1} = \text{anything}, Y_{1:t} = y_{1:t}) \\
&= P(X_t = x_t , X_{t-1} \in  \bigcup_{x_{t-1}} x_{t-1}, Y_{1:t} = y_{1:t}) \\
&= \sum_{x_{t-1}}P(X_t = x_t , X_{t-1} = x_{t-1}, Y_{1:t} = y_{1:t}).
\end{align*}
In the last line you are summing over all possible values of $X_{t-1}$. The set of all possible values is sometimes called the state space, or the support. You are allowed to do this because of the third axiom of probability functions/measures: countable additivity.
