PAYE stands for Pay As You Earn, and this is exactly what the system entails.
Any person who is employed must be registered under a PAYE scheme, which should ensure that the person pays the correct amount of income tax and national insurance.
Income tax is payable as a percentage of earnings above an 'allowance', which is set each year by the government, and runs from the 6th April to the 5th April the following year (a tax year).
If an allowance is not used, it is carried forward for use in the next month or week.
National Insurance, depending on the persons personal situation, is usually payable as a percentage of earnings above another 'allowance', which is set each year by the government.
This allowance is slightly different from the Income tax allowance, as it is a weekly allowance, and cannot be carried forward from week to week.