Reddit reviews The Vision of Ayn Rand: The Basic Principles of Objectivism
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The best option is almost certainly Leonard Peikoff's "Objectivism: The Philosophy of Ayn Rand" (aka OPAR), based on lectures Peikoff gave in the 70s with Rand's approval:
http://www.amazon.com/Objectivism-Philosophy-Ayn-Rand-Library/dp/0452011019/
The other major option is Nathaniel Branden's "The Vision of Ayn Rand", which is essentially a transcript of an earlier set of lectures Branden gave in the 60s, also with Rand's approval (and before Rand broke ties with Branden):
http://www.amazon.com/Vision-Ayn-Rand-Principles-Objectivism-ebook/dp/B00LV0FX2S/
Nathaniel Branden delivered a treatise in lecture form called The Basic Principles of Objectivism. It has now been transcribed to book form as The Vision of Ayn Rand. Both are quite good.
Lectures: https://thecultureofreasoncenter.com/the-basic-principles-of-objectivism.html
Kindle edition for the book: https://www.amazon.com/Vision-Ayn-Rand-Principles-Objectivism-ebook/dp/B00LV0FX2S/ref=sr_1_1?ie=UTF8&qid=1502550110&sr=8-1&keywords=the+vision+of+ayn+rand