Subjectivism

Subjectivism is the doctrine that reality, truth or values are determined by consciousness rather than discovered by it — that a thing is so because someone feels, wishes or decrees it to be so. Objectivism rejects it as one half of a false alternative, the other half being intrinsicism, and offers objectivity as the third possibility both of them omit.

“Subjectivism is the belief that reality is not a firm absolute, but a fluid, plastic, indeterminate realm which can be altered, in whole or in part, by the consciousness of the perceiver.”
~ Ayn Rand, "Who Is the Final Authority in Ethics?", The Voice of Reason

The root: the primacy of consciousness

Every version of subjectivism rests on a single metaphysical premise — that consciousness is prior to existence, so that what is depends on what is thought or felt. Objectivism holds the reverse: consciousness is the faculty of perceiving that which exists, and a faculty with no independent object to perceive would have nothing to be conscious of. See Primacy of Existence and Primacy of Consciousness.

The premise appears in several forms, which look unrelated until the root is identified:

Personal subjectivism — "true for me". Each consciousness makes its own reality.
Social subjectivism — truth is what the group, the culture or the consensus holds. This is the more common modern form, and it is not an improvement: a hundred consciousnesses have no more power over facts than one. See Collectivism.
Supernatural subjectivism — reality is the product of a consciousness outside it, whose decrees define what is true and good. See Mysticism and Faith.

In ethics

Applied to values, subjectivism holds that the good is whatever someone wants, prefers or approves of — that a value-judgment reports a feeling rather than identifying a relationship between a thing and the life of the person evaluating it. Its slogans are "who's to say?" and "it's better not to judge".

Objectivism's objection is not that this is unkind but that it is false, and that it makes ethics impossible in the process:

If the good is whatever one wants, no one can ever be mistaken about his own interests, and the entire subject has nothing left to investigate. See Value and Life.
Rand's term for this in practice is whim-worship: acting on desires whose source and consequences one has not examined. She treated it as a failure of egoism rather than an instance of it, since a whim is precisely not a judgment about what serves one's life.
Because it removes any standard by which to settle a disagreement, the only thing left to settle one is force. This is why Rand argued that subjectivism in morality issues, politically, in the rule that might makes right. See Initiation of Force and Statism.

In epistemology

Applied to knowledge, subjectivism holds that concepts are arbitrary constructs rather than identifications of real similarities — a position that reaches philosophy most often as nominalism. If classification is conventional, then no classification is more valid than another, and knowledge reduces to the manipulation of labels. See Concept and Objectivity.

The consistent form is skepticism: the claim that certainty is unavailable to man. Objectivism answers that this claim is itself asserted with certainty, and so belongs to the stolen concept family. See Certainty and Truth.

Common misunderstandings

That it means having preferences. Preferring vanilla to chocolate is not subjectivism. Holding that there is no fact of the matter about whether a diet will kill you is.
That the alternative is intrinsicism. Objectivism rejects both. Values are neither invented by a valuer nor inherent in objects apart from any valuer; they are relationships between a thing and the life of the person for whom it is a value.
That tolerance requires it. Refusing to condemn is not the same as having no basis for judgment. Objectivism holds that suspending moral judgment protects the guilty at the expense of the innocent.
That "subjective" and "personal" mean the same thing. A fact about one particular person — what he can eat, what work suits him — is objective and contextual, not subjective. See Context.

Discussion

References

Rand, Ayn. "Who Is the Final Authority in Ethics?", The Voice of Reason: Essays in Objectivist Thought.
Rand, Ayn. "What Is Capitalism?", Capitalism: The Unknown Ideal — the intrinsic, subjective and objective theories of value.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 4.