If your employee is under 65 their first £7475 per annum of income (from their state pension, private pension, investments, share dividends, employment and all other taxable sources) is free of tax.
If your employee is between 65 and 74 the first £9940 of their income is free of tax. If they're 75 or over that figure rises to £10,090.
So if, for example, your employee has already used up their tax-free allowance (because their pensions and other income already meet or exceed the relevant figure above), they must pay tax at 20% on every single penny that they earn from you. But if their other income doesn't take them above their personal allowance they only have to pay tax on that part of their income which takes them over that allowance.
Either way, HMRC will regard it as YOUR duty (under the PAYE rules) to ensure that the correct tax is deducted. (They won't look kindly upon a 'cash in hand' arrangement).
Further, if your employee has not reached the state pension age (but has, for example, retired on a private pension) and they earn more than £139 per week you must deduct 12% of their pay which is over £139, as National Insurance. You must also pay the employer's contribution from your own pocket. If the employee has reached the state pension age you must still continue deducting (and paying) National Insurance unless your employee has completed the relevant paperwork which exempts them (and you) from paying it.
Start here for information about PAYE:
http://www.hmrc.gov.uk/paye/index.htm
Chris