A certain number of shares of stock are set aside for you. When they "vest", that means you get them.

The vesting schedule varies, but probably the most common arrangement is that you get 1/4 of the shares per year for the next 4 years. Often you get nothing until you've been there for a year, then get a whole year's worth of stock in one big lump sum, and then get smaller amounts more frequently thereafter.

In the US, the stock is taxed as ordinary income on the day you get it. If its price goes up before you sell it, you also pay capital gains tax on the change in price that happened while you were holding it. (I sell all my stock immediately on general principle, so my capital gains taxes are tiny.)

Vest means you sign on you get 40 shares, but they become available over a delayed period. 5 vest in the first year 5 in the second 10 in the third 10 in the fourth

Vest means become available to sell or trade. You are also given stocks at bonus time. Encouraging you to stay around.