Earnings per share is the portion of a company's profit assigned to each outstanding share of its common stock. The Securities and Exchange Commission describes it, in defining the price-to-earnings ratio, as "the earnings for the past 12 months" divided by the number of common shares outstanding. More precisely, the numerator is net income reduced by any dividends owed to preferred shareholders, since those are not available to common shareholders, and the denominator is the number of common shares. The result is a per-share profit figure that makes companies of different sizes comparable on a like-for-like basis.
Earnings per share matters most as the bridge between a company's profits and its stock price. The price-to-earnings ratio, the most common valuation shorthand, is just the share price divided by earnings per share, so EPS is the profit half of that comparison. On its own, EPS says how much profit each share earned; paired with the price, it says how much a buyer is paying for each dollar of that profit.