How you are paid depends upon your type of employment and employer. Some employers will pay you on a weekly basis, others on a bi-weekly (every other week or twice a month) and still others on a monthly basis.
Most will pay by payroll check -- a check that clearly shows your salary for the pay period and what money has been taken out for taxes. Some employers may offer you the option of receiving an automatic deposit. This means that your paycheck is automatically deposited in your checking or savings account. Automatic deposits can be convenient, but be sure your employer is trustworthy, only has permission to make deposits (not withdrawals!) into your account (you can bring the paperwork you fill out for automatic deposits to your bank and ask them to review it), and check each deposit on your bank statement to make sure it is for the correct amount.
In general, employers should not be paying you by cash or personal check. If they are, they should be giving you a detailed statement showing how much money they have set aside for federal and state taxes. Usually it is an employer's responsibility to take out and pay your taxes directly to the government. Occasionally part-time workers, freelancers, or consultants will agree to pay their own taxes, but this often takes time and can cost more money, so before you agree to pay your own taxes (rather than have your employer pay), be sure to talk to a trusted financial advisor. The next question will tell you more about where your tax money is going.