What are we actually testing in the Augmented Dickey fuller test? 
I have been trying to understand the Augmented Dickey-Fuller test and have watched many lectures and videos about it, and I would say the video from ritvikmath is the clearest one.
He says that to test whether there is a unit root we do a t-test for the delta term (Yt-1 coefficient) and test it against the Dickey-Fuller distribution, and we do a regular t-test for the beta coefficients for the Yt-i terms, comparing against the regular t distribution.
What I don't get is that delta and beta coefficients are just sums of coefficients, for an AR(3) the delta term that we would test is ∂1+∂2+∂3-1 or a sum of 2 coefficients minus 1. However we don't have any actual terms for those coefficients, so what are we testing?

That is the sum of coefficients minus 1 which represents delta in rikvitmath's video, but how do we do a t test for coefficients with no assigned values?

This is the equation to find the t value but pi in this case is just a representation of the summation of coefficients with no value, so how does this calculation work?
Furthermore, the regular t test for the beta value states βi. but i stands for the order of the auto-regressive model, how does a computer determine how many lags there are, or what order AR(p) the time series is.

I looked back at my time series notes in response to your question. I hope the points below on unit root tests are made relatively clear.
In particular, I always found it best to understand the augmented Dickey-Fuller (ADF) by first understanding the Dickey-Fuller (DF) test:
The Dickey-Fuller Test:
Suppose that we have an AR(1) unit root process:
$$y_t = \phi_1 y_{t-1} + \epsilon_t, \qquad \epsilon_t \sim(0,\sigma^2) $$
The DF proceeds by taking advantage of the fact that a first-differenced random walk/unit root process is stationary.
In other words, the first-differenced coefficient $\psi_1$ of a random walk should be zero; given the non-first differenced random walk coefficient is 1.
The null and alternative hypothesis of the DF test are thus expressed:
\begin{equation}
\begin{split}
H_0: & \psi_1 = 0 \\
H_A: & \psi_1 <1
\end{split}
\end{equation}
Upon which we test the first-differenced process:
\begin{equation}
 \begin{split}
  y_t - y_{t-1} & = \phi_1 y_{t-1} - y_{t-1} + \epsilon_t \\ 
  \Delta y_t & = (\phi_1 - 1)y_{t-1} + \epsilon_t\\ 
  \Delta y_t & = \psi y_{t-1} + \epsilon_t   
 \end{split}
\end{equation}
The Augmented Dickey-Fuller Test:
Note that upon closer inspection of the the standard DF test, we required that $\epsilon_t$ is is a stationary white noise error term.
\begin{equation}
 \Delta y_t = \psi y_{t-1} + \epsilon_t 
\end{equation}
Yet, the validity of the DF test is clearly undermined in the circumstances where the assumption regarding the error does not hold.
The ADF test overcomes this problem by placing lags of the dependent variable on the right-hand side to ensure that the error is zero.
\begin{equation}
  \Delta y_t  = \alpha_0 + \psi_1 y_{t-i} + \gamma t \sum_{i=1}^{p-1} \delta_i\Delta y_{t-i} + \epsilon_t 
\end{equation}
\begin{equation}
\begin{split}
H_0: & \psi_1  = 0\\
H_A: & \psi_1 <1
\end{split}
\end{equation}
The additional terms in the model such as $\alpha_0$ and $t$ express any additional deterministic terms required in the specification of the model.
Overall, the ADF test generates a p-value where we would reject the null (unit root) if within the 0.05 rejection region.
