Options aren't equity, they're only the option to buy equity at a specified price. Vesting just means you can actually buy the shares at the set strike pice.
For example, you may join a company and be given options to buy 10,000 shares at $5 each with a 2 year vesting schedule. They may begin vesting immediately, meaning you can buy 1/24th of the total options each month (or 614 shares). Its also common for a delay up front where no options vest until you've been with the company for say 6 or 12 months.
Until an option vests you don't own anything. Once it vests, you still have to buy the shares by exercising the option at the $5 per share price. When you leave, most companies have a deadline on the scale of a few months where you have to either buy all vested shares or forfeit them and lose the stock options.
