What are the differences between Fire Emblem Fates: Birthright, Fire Emblem Fates: Conquest, and Fire Emblem Fates: Revelation?

To add onto Marth's answer:

All games are the same up until Chapter 6, when you are forced to make a choice. Conquest aligns you with Nohr, Birthright with Hoshido, but Revelations aligns you with neither.

Revelations is a blend of Conquest and Birthright, both story-wise, and gameplay-wise. You will be able to recruit almost every character available in Conquest in Birthright, which will unlock extra supports and 2 marriages per version-exclusive character.

In terms of gameplay, the campaign is less difficult than Conquest, but more rigorous than Birthright. From personal experience, I believe the difficulty of Revelations is closer to Conquest, but you are able to train your units on previously completed maps, similar to Birthright.

In your Castle, you can have buildings from both games (i.e. you could produce both ruby and coral, or buy tomes as well as staves).

It is worth noting, however, that it is intended for the player to at least play either Birthright or Conquest before playing Revelations.