Two friends, Joe and Anna, start a company. They're both going to be working on it really hard, full time. They decide to split the ownership of the company 50-50.
After three weeks, Anna quits. She can't stand Joe. He's actually getting beyond annoying. Joe continues working on the business for five years. It becomes HUGE. It's like Facebook, only bigger.
The company goes public. Anna shows up with her stock. "OK, Joe," she says, "give me my 50% of the shares now. Thanks for making me a multi-trillionaire!"
Joe says, "That's not fair. You quit after three weeks. I've been working on this day and night for five years while you went back to graduate school. And the word is "trillionairess."
To prevent this kind of problem, you institute vesting for all founders and employees of a business. Instead of getting their share of the stock outright, they have to earn it (vest it) by continuing to work for the company for a period of time. That is called vesting.
In a typical startup shares vest over four or five years. Often there is a "cliff" -- you don't vest ANY shares for the first year.
Technically, the way this is usually implemented is as follows:
Basically if you're given stock options at a company then you can't just cash in and leave. You have to wait for them to 'vest', essentially mature. After they have vested you can cash in. There may be different vesting schedules, for instance 20% in year 1 and the remaining 80% in year 2.
Small entrepreneurial companies usually offer grants of common stock or positions in an employee stock option plan to employees and other key participants such as contractors, board members, and major vendors. To make the reward commensurate with the extent of contribution, encourage loyalty, and avoid spreading ownership widely among former participants, these grants are usually subject to vesting arrangements.Wikipedia: Startup Vesting
Common stock grants are similar in function but the mechanism is different. An employee, typically a company founder, purchases stock in the company at nominal price shortly after the company is formed. The company retains a repurchase right to buy the stock back at the same price should the employee leave. The repurchase right diminishes over time so that the company eventually has no right to repurchase the stock, i.e. the stock becomes fully vested.
The problem I see with "classical" vesting is that it replaces "fixed equity carved in stone" with "fixed vesting schedule/rule carved in stone". The first line in Wikipedia's article says it all:
In law, vesting is to give an immediately secured right of present or future enjoyment.So you are entering your startup and want something "immediately secured"! Good luck :)
I know, it does not answer the exact question (for which other answers already did superb job), but I believe it is something to think about when your interest in the topic comes from real life as opposed to theoretical terminology question.
I my other answer I wrote up some more thoughts and information that does not answer the terminology question but can be helpful to deal with the root problem, for which vesting is just one of the tools - finding a fair way of equity allocation.