In hypothesis testing, we set up a rejection area for rejecting $H_0$ in favor of $H_1$ with $\alpha$. I don't understand why type I error (rejecting $H_0$, when $H_0$ is actually true) is the area that we choose to reject $H_0$. Why does it makes senses that we reject something from error rates?
a type I error is the mistaken rejection of an actually true null hypothesis
$\alpha$ has the same value as type 1 error but you can distinguish them by going thru the following story.
In your hypothesis test of recovery rate of a drug, you first assume your $H_0$ is correct, which means the drug gives you the same recovery rate as not using the drug. In this case, you assume the recovery rate distribution of the drug is the same as the distribution of not using drug.
Then you calculate the average recovery rate of patients using the drug, and find a value $r$. Then you look at where $r$ is in the $H_0$ distribution. and you calculate the $\alpha$ by summing the area under the distribution $>=r$, so this $\alpha$ value is actually the total probability of observing a recovery rate $>=r$ assuming that $H_0$ is correct. Let's say this value is 0.021, meaning that if $H_0$ is true, you have 2.1% chance observing $r$ or greater than $r$. It is quite unlikely judging from the size of the value, but it is NOT IMPOSSIBLE.
Then you need to decide whether you want to reject $H_0$, you can reject it like a dictator or you can reject it by saying that "OK, now $\alpha$ is smaller than a threshold which I decided to be 0.05, so let's reject $H_0$.
Now we finally introduce "type 1 error".
We can reject $H_0$ because we think the chance of observing $r$ is small, so it is merely not likely, but not impossible. Therefore we can be wrong!
If we are wrong, then $H_0$ should actually be true, and how likely it is that we are wrong (error)? It is how likely it is for us to observe $r$ or greater than $r$, which is $\alpha$, which is 0.021. Therefore the error value has the same value of $\alpha$, and this error is named as type 1 error.