I did the Ancestry DNA an had a match of 1671 to a person I did not know. My sister did the test and matched him at 1251. My father believes he has a son he does not know and the stories seem to add up. My question is, from everything I have read my sister's numbers should be higher in order for him to be a half sibling.
I am assuming that you and your sister match as expected, as either paternal half-siblings or full siblings.
According to the data that Blaine Bettinger has gathered through his Shared cM Project, yes, the shared DNA between your sister and your potential new half-brother falls outside the range for a half-brother: Shared cM Project for 1251 cM. However, if you look at that page, it lists a 28.26% probability for some additional potential relationships, including half-sibling. This potential relationship comes with the following note:
this relationship has a positive probability for 1251cM in thednageek's table of probabilities, but falls outside the bounds of the recorded cM range (99th percentile)
You can read more about this in thednageek's post The Limits of Predicting Relationships Using DNA.
I think the best next step would be to ask your father to do a DNA test. Then you'd get a solid answer about how your father and this unknown person are related.