Any income you earn, you must give a portion to the government in taxes but how do you determine how much? By January 31th, your boss must send you a W-2 which includes how much you earned working for them, how much was taken out and sent to the federal government, the state government, social security and medicare. Then by February 15th any bank or brokerage firm must send you a 1099, which is any interest or capital gains you receive over the year. Be careful because some companies will only send out a 1099 if you make more than $10 but you still have to report the income to the IRS even if you do not receive a W-2 or 1099. So once you have your W-2 and 1099 (if any) what do you do?
If you go to the free file section on the irs website (http://www.irs.gov/efile/article/0,,id=118986,00.html), you can find companies that allow you to use their programs for free if you earn under $57,000. My favorite is turbo tax. It will walk you through your taxes and allows you to file online. You do have to pay for the state but you can print out your federal and use it to do your state taxes and avoid paying for anything.
Remember if you are a student, wait for your 1098-T Tuition Statement which says how much you spent in tuition and fees before you file. You can either get a tax deduction (see http://frugalstudents.blogspot.com/2011/07/higher-education-deductions.html for more information about the deduction) or tax credit (see http://frugalstudents.blogspot.com/2011/06/tax-credits-for-education.html for more information about the credit) for your tuition and fees. Turbo tax or most other tax programs will help you determine which will save you the most money but normally, the credit is best.
Feel free to leave any questions about taxes in the comments. I can answer tax questions for federal, California and New York taxes but I am not a CPA or EA.