Totally depends on the state. In Ireland you don't pay anything when the stocks are granted to you (you're promised X stocks in Y months), you pay income tax (PAYE) on them when they vest (you receive the stocks) and then you pay capital gain tax (CGT) on the gains (sell price - vest price) when you sell them.

Example: you're granted 1 share today to vest in 1 year. In 1 year the share is worth 100, it vests and you pay PAYE in this (it can either come from your salary, or by selling the stock). If the stock rise to 150 and you sell it after, you pay CGT on 50.