The chair is an abstract role played in the course of a meeting, as well as the physical chair that symbolizes that role. The chair, incarnated as the person who sits in it, directs the flow of the discussion and decision-making process, typically according to some set of rules. The chair determines who has the floor, and usually people who don't have the floor must address their remarks to the chair or be considered out of order.
If the person who normally sits in the chair is a man, few people will complain if you call him a chairman. For some reason, however, the term chairwoman is frowned upon. Sometimes people take offense at the extra syllable (which feels cumbersome), and sometimes they take offense at the use of the suffix "-woman", which can be taken to indicate that a woman holding a job is somehow different and special compared to the supposedly natural state of a man holding the job. The term chairperson, which is no less awkward but at least more generic and potentially less offensive, fills this role instead. (But the generic nature of the term still doesn't mean men get called "chairperson" all that often in practice. So I just don't get it, really. Would it be OK if I just said "chairwomyn"?)
Sometimes people try to skip all that guff and call the person the "chair" instead. As silly an image as this is, it does avoid both the gender politics and awkwardness of the other terms. Nevertheless, it is historically incorrect and generally considered poor usage, as though you were referring to Queen Elizabeth II as the Monarchy instead of the Monarch.
So, to summarize:
Right: "The chair called the meeting to order."
Wrong: "She is the chairman of that committee."
Wrong: "She is the chair of that committee."
Accurate, but cumbersome and politically incorrect: "She is the chairwoman of that committee."
Cumbersome, but accurate and politically correct: "She is the chairperson of that committee."
You just can't win.