I teach ESL and each book that I read gives different information. Could you please help me out on this one? Thanks.

Is this like Polls, mostly for fun ??? I'll answer, but, can you tell me ( or add in the details section, what ESL is ???)

A rail is like on railways; transportation material for trains.

A railing is like in a building, or entrance to/exit from one.

That is the simplest answer.

But, am really curious: what is ESL ??

I would claim that the full name of the object is "hand railing". "Hand rail", and "railing" are all common abbreviated references. "Rail", while it may "work" (as people understand the context), has other associations that make it a less effective abbreviation.

Rail has been defined in many ways, it depends on the situation.
1.Rail = rod
e.g.Hold onto the rail so that you don't fall.

2.rail= train
e.g.Environmentalists argue that more goods should be transported by rail.

3.rail= complain
e.g.He railed against/at the injustices of the system.

4.railing = wooden/metal fence
e.g.Tourists pressed their faces against the palace railings.