Three averages, three different questions
Every dataset can be reduced to a single "typical value," and the three classic candidates each answer a subtly different question.
The mean (arithmetic average) is the balance point: the value you would assign every observation so the total stays the same. It uses every data point, which makes it efficient when the data are clean — but that same completeness makes it fragile. One apartment listing at ten times the market rate moves the neighbourhood mean by hundreds of dollars while the median barely stirs.
The median is the middle value: half the observations sit above, half below. It is the statistic you want for skewed quantities — incomes, home prices, response times — because it describes what a typical case actually experiences. The median house price is the price of the house in the middle of the market, not a mathematical phantom inflated by three mansions.
The mode is the most frequent value and the only average that applies to nominal data (categories have no middle or balance point). It answers "what is most common," which is why it is the natural summary for survey pick-lists and the only average you can take of *yes/no* answers.
Weighted variants fix mismatched importance: a grade-point average weights each course by credits, a portfolio return weights each holding by its share. The trimmed mean deletes a stated percentage from each tail before averaging — a deliberate compromise that keeps most of the data's efficiency while discarding the most extreme contamination.