Hello there, welcome to WuxiaSociety! They all refer to factions in general.
Here’s how I translate them on WuxiaSociety:
帮 bāng – Guild
A group of people brought together by a common interest or profession. Such as the Beggars’ Guild.
派 pài – Sect/Order
A school of thought. In wuxia, it is very often referring to schools that believes in training in martial arts (学派) or cultivating oneself in a certain way (教派). I use Sect when the school of thought is martial arts-based, and Order when it is religion-based. Such as Huashan Sect or Shaolin Order and Wudang Order.
教 jiào – Order
A religious or quasi-religious order order, similar to the second type of pai I mentioned above. They combine martial arts with spiritual or philosophical teachings. Some examples besides the ones mentioned above are Ancient Tomb Order and Sun Moon Order.
I hope this answers your questions.