A man decides to test if a coin is fair, or biased towards heads. He flips the coin 1000 times, and it comes up heads 500 times. His strategy is as follows. All we have to do is assume the coin is fair, and calculate the probability, under that assumption, that we would have gotten the result we got. If that probability is very low, we conclude that if the coin were fair, this outcome would be exceedingly unlikely, and therefore, the coin is probably not fair.
The man computes the probability of a fair coin coming up heads 500 times, and finds it comes to $3.05\cdot10^{-151}$. So he concludes the coin is biased towards heads.
In the language of statistical tests: he's using a test that consists in concluding the coin is biased iff it comes up with exactly $500$ heads. We've shown that the probability of type 1 error for this test is essentially zero.
Why is this reasoning not valid?
The idea of this test is "my particular data is very unlikely under the null hypothesis so I reject the null". The basic flaw in it is that any particular data is very unlikely under the null hypothesis, so you'll reject the null regardless of what data you actually get. This "paradox" is more dramatic with continuous distributions in which case the probability of observing any particular data is zero.
In the usual framework, this is taken care of by instead measuring the probability that data at least as favorable towards the alternative hypothesis as the given data would be seen under the null hypothesis. So in your situation you would consider the probability that $|X-500| \geq |x-500|$ where $X$ is the random number of heads in a flip and $x$ is the measured number of heads. The smaller this probability, the less reasonable your data makes the null appear.
In even more bland statistics jargon: a test like yours always commits a type 2 error if the null holds and never commits a type 1 error. Statisticians at a cultural level are more concerned about type 2 errors than type 1 errors, which in more colloquial language means they would rather say they don't know than say something false. You can set your priorities the other way around if you want, though you will need to be a bit less extreme about it to avoid creating weird situations like this one where you essentially deny any claim that anyone makes about anything.