Is there a particular name for this kind of "poster"? Is this called a poster?
Yes, that is a poster. I would call it a poster with tear-off tabs. I think Microsoft Word even has a template for this type of poster.
Update: Microsoft does offer a Word template and calls it a poster with tear-off tabs.
I'd call it a flyer with pull tabs. The more common phrase is flyer with tear-off tabs, perhaps because another type of "pull tab" can be used to open a can. In this case, though, "pull tab" has its origins in gambling, where you'd pull a tab on a ticket to reveal if you had won a prize.
It's worth noting that flyer can also be spelled flier.