Allora translates more English words:
- then: If it rains, then I'll stay at home [Se piove, allora starò a casa]
- (back) then: It was 1950. I was a kid back then [Era il 1950. Allora ero un bambino]
- so: I couldn't find my keys, so I knocked at the door [Non riuscivo a trovare le chiavi, allora ho bussato alla porta]
In this last case, you could also swap allora with così, as it is a synonym: [Non riuscivo a trovare le chiavi, così ho bussato alla porta]
Poi can only be used as a "temporal linker". It links two actions together that don't have the "cause & effect" meaning that allora has.
- after that, then: I went to the movies, then I went to a pub [Sono andato al cinema, poi sono andato a un pub]
Hope this helps!