The quick answer
Use the mean when your data doesn't have extreme outliers and every value should count equally toward the average. Use the median when a handful of extreme values would distort the mean — income, house prices, and salaries are the classic examples. Use the mode when you want to know which single value or category occurs most often, which is the only question mean and median don't answer at all.
A worked example where mean and median disagree sharply
Six households report monthly incomes of ₹20,000, ₹22,000, ₹22,000, ₹23,000, ₹21,000, and ₹2,50,000 (one household earns far more than the rest). The mean is ₹59,666.67 — pulled up almost threefold by the single outlier, and not a figure that describes any of the other five households. The median is ₹22,000, and the mode is also ₹22,000 (it appears twice) — both land squarely in the range where five of the six households actually sit.
Why this happens
Mean uses every value in its calculation, so one unusually large (or small) number pulls it directly. Median only cares about which value sits in the middle position once the list is sorted, so an extreme value at either end has no effect on it beyond its rank. This is exactly why median household income, not mean household income, is the figure typically reported in economic statistics — see the companion article on how to calculate mean, median, and mode for the mechanics of finding each one.
When mode is the right choice
Mode is the only measure of the three that works on non-numeric categories — the most common shoe size sold, the most frequent survey response, the most common blood type in a population. For numeric data with mostly-unique values, mode is often unhelpful (there may be no repeated value at all) — the companion article on why a list can have no mode or multiple modes covers that case directly.