Where do ideas come from? What are concepts? What is the nature of moral principles? Do mathematical entities exist beyond the reality we can perceive with our senses? Where can we find the essence of an entity? Do essences exist, or are they only in our mind? These are some of the questions that philosophers have been asking since the beginning of recorded civilization. These are also some of the questions we ask when we are kids, but then we stop asking when we realize that these questions are hard and we find other things to keep us occupied.
Ayn Rand came up with an answer to this, and in reaching her answer she came up with a very useful categorization of all the possible answers to these questions. This categorization, which partitions the space of possible answers (i.e. separates it into a set of categories that are mutually exclusive and jointly exhaustive), is the distinction between regarding concepts as intrinsic, subjective, or objective.
Intrincisism is the position that concepts are intrinsic, that they exist in reality independent of the mind of man. This is the position that Plato held, when he postulated that this world is only an imperfect reflection of another world, the world of the perfect forms. This is the position that religious people hold regarding moral values, they exist independent of man in the mind of God, and they are revealed to man through intuition. This is also the position that some mathematicians like Roger Penrose hold regarding mathematical concepts, that these concepts exist independent of man, and when we learn a new mathematical truth all we are doing is discovering one of these preexisting concepts.
Subjectivism is the position that concepts are subjective, only in man's mind, and that they are arbitrary conventions that we created in order to communicate with other people but with no reference to reality. This is the position of some modern philosophers, like Nietzche who rejected moral principles, or the logical positivists who reject the existence of anything that cannot be measured. Some modern philosophical movements such as Marxism, Nazism, multiculturalism, and feminism, believe that the conceptual structures people create in their minds depend on their social class, their race, their culture, or their gender, and that none of these have any referents in the real world, so none of these can be considered more valid than any other.
Objectivism is the position that concepts are objective, that is, they exist in the relationship between reality and man's mind. The mind observes the world and creates concepts as a way to understand reality, based on abstracting and inducing from direct observation. So concepts cannot exist in reality without a mind, or in a mind without reality, you need both of these to create a concept. This is the position Ayn Rand held in her theory of concepts, and that position appears everywhere on her philosophy, from epistemology, to ethics, politics and aesthetics.
To illustrate the profound difference between the three approaches let's look at ethics. Intrincisists hold that there is one true set of moral principles that we have to somehow learn through divine revelation or intuition. Subjectivists hold that our morality is an arbitrary product of our culture, part of a social contract that could have been otherwise. Objectivists hold that morality is necessitated from the nature of man, and that its principles, like the principles of physics, must be discovered by observing reality. For an intrincisist, morality is a set of rules imposed by a deity or a noumenal world. For a subjectivist, morality is a set of rules imposed by society. For an objectivist, morality is a set of principles one must follow in order to achieve happiness in life. Because of this, the morality of Objectivism is rational egoism, which is very different from the altruism and the irrational egoism espoused by intrincisists and subjectivists.
Another example is mathematics. Intrincisists think that mathematics exists somewhere beyond reality, in the world of forms, and that one must discover them. They have no problem explaining why reality seems to follow mathematical rules since to them, reality is just an imperfect copy of that world of forms. Subjectivists think that mathematics are just an arbitrary game of symbol manipulation people play. So they are at a loss when trying to understand why mathematics can explain and predict the real world. Objectivists think that mathematics are part of the form in which a conceptual consciousness perceives and abstracts reality. So they have no existence without the mind, but they exist as a conceptualization and abstraction of reality, so there is no mystery as to why they explain and predict the real world. They do so because they are abstracted from relationships observed in reality.
In science, the method of Objectivism is induction of principles based on observation and experiment, the method of Intrincisism is thought without experiment (rationalism), and the method of subjectivism is experiment without conceptualization (empiricism, and skepticism).
In politics, the left is a form of subjectivism, and the religious right is a form of intrincisism. Objectivism holds that the recognition of individual rights is the necessary precondition for a rational being to be able to gain from living with other rational beings, and that the rights of man can be objectively derived from the nature of man. Therefore, Objectivism advocates a limited government whose sole purpose is to protect those rights. The left believes that there is no such thing as natural rights, and that society can postulate new rights, or remove some rights if the majority feels that this is correct. The religious right believes that rights are given by God, and that government should be based on religious principles. Therefore they reject any right that conflicts with their teachings.
As you can see, whether one regards concepts as intrinsic, subjective, or objective, makes a huge difference in the subsequent development of one's ideas about many aspects of reality. Therefore it is very important to understand the basis of the ideas that one holds, and to identify if one's ideology comes from an intrinsic, a subjective or an objective basis.