What it means
A model learns the patterns present in its training data, including the ones reflecting historical discrimination. Trained on past hiring decisions, it learns who was hired before; trained on internet text, it absorbs the associations that text carries.
The subtle mechanism is proxies. Removing a protected attribute does not remove the bias, because other fields correlate with it — postal code with race, school or career-gap patterns with gender. A model can reconstruct what you deleted from what you kept.
Bias is measurable but not single-valued: there are several mathematical definitions of fairness, and they are provably incompatible with each other except in degenerate cases. So "is this model fair" has no purely technical answer — choosing a fairness definition is a normative decision that has to be made explicitly.
Why it matters
In hiring, lending, housing, insurance and criminal justice, biased outputs are illegal, not merely regrettable, and existing anti-discrimination law applies regardless of whether a model made the decision. "The algorithm did it" is not a defense in any jurisdiction.
In practice
For any consequential decision, test outcomes across groups rather than auditing inputs — proxies make input auditing insufficient. Pick and document your fairness criterion, since you cannot satisfy all of them.