I know, by definition, what p-value and type 1 error mean. However, I have hard time relating those two concepts in rejecting a null hypothesis.

Below are my understanding about P-value and Type 1 Error

1)A p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. (from wiki)

2)A type 1 error incorrectly rejects a true null hypothesis. That is, reject the Null when in fact the Null is true.

My question is - Why do we reject the null hypothesis when p-value < type 1 error? What are some intuitions behind it? What am I missing..? After learning 1 year of statistics, I still have no idea how this works..

Thanks.