A company's earnings per share (EPS) is their net profit divided by the number of common shares outstanding.
Suppose a company had 1B USD net profit one year with 500M shares outstanding. It's EPS would be 2 USD per share that year - the amount of profit earned per share of stock. The share price divided by the EPS is the P/E ratio. A so-called forward EPS may be computed using projected future earnings. Diluted EPS and adjusted diluted EPS are other metrics that account for potential share dilution (e.g. bonds convertible to common stock) and unrealized gains or losses.
Earnings per share in Bankrate's glossary
Investopedia's article on earnings per share
What earnings-per-share tells investors