I attend ''a speakers' club'' or ''a speakers club'' or ''a speaker's club''.
Which one is correct and why? What do others mean?
Any one of them could be correct and their meaning is very similar.
I attend a speakers' club
Here the club belongs or pertains to numerous speakers. My feeling is that this is the most common case.
I attend a speakers club
Here "speakers" is an adjective. "Speakers" is a type of club--one filled with speakers. It shifts the view away from possession to description. Normally this is done only with common cases, where it has become the norm to refer to something like a club this way. That is, if you just use it yourself without seeing someone else do it first, this is likely to be an error. Note that it could also be a "speaker club."
I attend a speaker's club
Grammatically this says that the club pertains or belongs to one speaker. This isn't super common. However, anyone who attends the club may consider themselves to be that speaker, so there are examples of this in usage.