TL;DR There's no real difference in how RSUs are taxed and how other stocks are taxed. It's just a matter of logistics. You can think of it as your company pays you a bonus which you are required to use to purchase company stock. You are taxed on the bonus, and then you buy stock. Later, you pay tax on any capital gain realized once you sell the stock.
There are two taxable events: one when you receive the stock, and one when you sell the stock. Like any other stock you hold, you are never taxed on an unrealized gain just because the stock price has gone up since you acquired the stock.
When you receive the stock, you immediately realize ordinary income (OI) based on the fair market value of the stock at that time. Say you receive 100 shares of stock that is trading at $50/share. This is equivalent to $5,000 in income, on which you are taxed.
However, you don't actually get all 100 shares. Usually, some of the shares are sold immediately and withheld, just like your gross pay may be $5000/month but your net pay is somewhat lower. You might receive, say 67 shares, with 33 shares being sold to cover the taxes due on the OI.
The OI and the withholding should be reported on your W2, just like your regular salary, so you don't have to do any real work in order to make sure those taxes are paid; it's taken care of for you.
Note that since you can't sell a fraction of a share, 32 shares may not have raised enough money to cover the taxes, but 33 shares raises more than you really needed. This excess money (called a "residual tax") is usually returned to you in your regular paycheck within a couple of pay cycles after the RSUs vest.
Later, when you sell the 67 shares, you may realize an additional capital gain, which is the difference between what you "bought" the stock at ($50/share) and what you sell it for (say, $52). Your capital gain is 67 * 2 = $134, taxable as appropriate based on how long you held the stock before selling and what state you live in.