The phrase "objective reality" means that reality exists independent of our minds. The description "objective" doesn't make a lot of sense on its own, but it does in comparison to the competing theory of the relationship between consciousness and existence.
The Primacy of Consciousness is the view, which Objectivism rejects, that reality is a product of consciousness. In this view, the world is a product of our minds, or the mind of some other being (god). It doesn't exist in itself. It exists as a "figment of your imagination". Like a concept or a memory, reality is said to exist as only a part of your mind. That means that you can change reality by changing your mind. Or another possibility is that you gain knowledge of reality by understanding yourself, or by introspecting. In either case, reality is said to not have it's own existence, but be just a product of your mind. If your mind ever stopped, like if you died, the world would die with you.
The Primacy of Existence, which is the Objectivist view, states that reality exists independent of our minds. If we want something to happen, we can't just wish it to happen. We have to act. If we want to gain knowledge about the world, we have to look out at the world and reason with what we see. If your mind were to stop functioning as in death, reality would stay the same. This is where the adjective "objective" comes into play in "objective reality". It means that reality is not just a part of our minds, but exists as some outside fact that we can reference.
There are many variations of the Primacy of Consciousness. One view would hold that your mind is the only mind that exists, and everyone else is a product of your imagination. That's a bit of a god-complex. Another view is that many minds exist, but we're ultimately unconnected to them since there is no reality. Still another view is that there are multiple minds, and that reality is a kind of group average of them. Each person has their own idea of what reality should be, and somehow they're all resolved in a common hallucination. You could spend hours coming up with other interesting and entirely useless alternatives.
The Primacy of Existence acts to remind us that the world is out there, and we have to act accordingly. In epistemology, it means that we have to look to reality to understand it. If we're mistaken about the nature of reality, then our actions won't work. Reality exists in a particular way, and if we don't grasp it, we lose. In ethics, it highlights the fact that you can't just wish for something, and you have to follow facts in order to get a desired result. If you act inappropriately, you will suffer. If you grasp reality and act properly, you'll live the good life. Some methods and actions are better than others when trying to achieve a goal. Some are just wrong, and cannot produce it.
The Primacy of Existence is there to keep clear the relationship between our minds and reality.