Independent contractors don't withhold tax; they estimate it. Because you don't have an employer to take out money from your paycheck, you have to estimate what you owe and send some of it to the IRS through the year. Estimated tax payments for the year usually fall due on April 15, June 15, September 15 and then January 15 of the following year.
If self-employment only brings in a couple of hundred a year, estimated tax is not an issue. You don't have to worry unless you expect to owe least $1,000 for the year. If you also have a day job, that means you expect to owe $1,000 on top of whatever is withheld from your regular paycheck. What's more, if you think your withholding will cover 90 percent of your tax bill, you don't have to pay estimated tax, either, even if you will owe more than $1,000. On the other hand, if you prefer to pay as you go, you always have the option send in estimated payments, even if it's not required -- and you'll avoid having to pay it all at once when you file your taxes.
IRS form 1040-ES is the one to use to figure out your taxes. It includes worksheets for calculating income tax, and for figuring out self-employment tax. This is the contractor's equivalent to the FICA taxes employees pay for Social Security and Medicare. You owe it if you earn more than $400 this year. If your self-employment income is steady, you can figure estimated tax based on what you expect to earn for the whole year, and you can send one quarter of that to the IRS each pay date. If your income fluctuates, you can do fresh calculations each time.
If your contracting doesn't turn a profit until, say, July, you don't have to make estimated tax payments in April or June. When you do calculate the tax, keep in mind any other income you have. Form 1040-ES uses your total adjusted gross income, not just your self-employment earnings, as the basis for estimated tax. If you file a joint return, you have to factor in your spouse's income and withholding too.
If you live in a state that charges income tax, you may owe estimated state income tax as well as federal. The payment dates and general principles are the same, but the figures probably won't be. Your state may not have the same tax rate, or use the same deductions and exemptions as the federal government, so you can't simply repeat your calculations and get it right. Fortunately, states typically provide you a similar worksheet to calculate your estimated payments. For example, California residents use a worksheet provided with Form 540-ES.