There are several problems with Ayn Rand.
One is that her ideas were not particularly original, and not particularly well-expressed. Self-centered philosophies are far from new, and actually are pretty well-trod ground, but her work does little to address already-existing critiques and, as literature, is not particularly good (her characters tend to be one-dimensional, plots lack good development/tension/resolution, etc.).
Another is that she has become a frankly cult-like figure, with people approaching her work the wrong way around: rather than "this statement is correct, and Ayn Rand said it", too often there is a seeming attitude of "this statement is correct because Ayn Rand said it". The Objectivist movement (people who follow her work and philosophy) is particularly infamous for this, having an established history of venerating her and doing some rather extreme turns when she was alive and particular people from her circle fell out of her favor.
Finally, most of her work is easy to critique with only very basic reasoning/argumentative skills, despite presenting itself as a solid, rationally-justified framework. More realistically, Rand's philosophy consists of appeals to emotional responses, based on the idea of self-evaluation of one's own greatness and the notion that this greatness exists more or less in a vacuum (one of the famous examples is "going Galt", wherein all of the great people who produce value simply retreat and form their own separate society, to spite the "parasites" who "leeched" off their work).
To continue with the fiction theme, one of Heinlein's stories ("The Roads Must Roll", 1940) anticipated and harshly criticized the type of philosophy Rand ended up promoting. One of the asides there is to a philosophy of "Functionalism"; the founder of the philosophy advocates evaluating people -- and giving them power and prestige -- based on what "function" they can perform, and how valuable it is to society.
The result is large numbers of people who do not really make any unusually-significant contributions, but who all come to the conclusion that whatever they do is the one truly indispensable thing, and if they stopped doing it the whole society would fall apart, so they should be given more power or prestige over others in recognition. As Heinlein puts it, "With so many different functions actually indispensible, such self-persuasion was easy." Heinlein also offers a description of the founder of "Functionalism" which critiques the philosophy and in many ways critiques Rand's later work as well, when he says:
The complete interdependence of modern economic life seems to have escaped him entirely.
(see Wikipedia for a summary of the story, or look it up -- it's been republished/anthologized: http://en.wikipedia.org/wiki/The_Roads_Must_Roll)