Most payroll systems are set up to withhold taxes on bonuses at specific, recipient-designated rates. (There is usally a default rate of 20 or 25%, but you can ask to have it changed.) Depending on your state, there is usually a standard rate for that as well. Then you have FICA withheld, too. That is the withholding portion.
As for taxes,taxes withheld from your normal salary use your gross income per period, the number of exemptions you declare, and withhold enough so that you would ideally break even at the end of the year. Bonus dollars do not figure into that. Our tax system is progressive. So the more money you make, the higher tax bracket you end up in. So, if at the end of the year you make taxable income of $xx,xxx, your total tax burden may only be 13% of your taxable income (for instance, if your tax liability is $13,000 on $100,000 of taxable income) but that rate is calculated by applying different percentages to the different brackets of income. Where you end up is the rate that any additional dollars will be paying taxes on. If you end up in the 28% tax bracket and you get a $1,000 bonus, you will owe an additional $280 in federal taxes (which will change your average tax rate to 13,280/101,000, or 13.15%). SO what you want is to have that $280 withheld.
So, look at the actual bonus check and see what percentage was withheld for federal income taxes. Look at your tax return and see what tax rate your taxable income ends up in. (There's a rate schedule, you might have to go to IRS.gov to find it. It will say something like if your taxable income is greater than this, you will pay $xx plus 28% for income over $xx,xxx.) This does NOT mean that your tax liability is 28% of your income!!!! It just means that at this point, every additional dollar you earn is taxed at this rate. Once you know your tax rate, you should be able to request that you have that percentage withheld from your bonus checks. Of course, there are dozens of other ways to play with your withholding to get it right, but this is probably the easiest.
Remember, the total taxes you pay is determined only by your tax return at the end of the year. What is withheld from your paycheck is just an estimate and can be adjusted to what works best for you.