A tax code is an instruction to payroll. It tells whoever pays you how much income can be paid free of tax, or which rate table to use, so the right amount of income tax comes off each payslip. You hold one for each job or pension, according to GOV.UK.
In the UK the number is your tax-free income with the last digit removed, and the letter says how it was reached. The common code 1257L means the standard personal allowance of £12,570, as HMRC explains. If you had a £2,000 taxable benefit, the allowance would fall to £10,570 and the code to 1057L. M and N codes show that a slice of allowance has been moved between spouses under the marriage allowance.
Normally payroll works cumulatively: it compares your pay so far this tax year with the allowance so far and taxes the difference, which corrects an earlier overpayment on the next payday. Emergency codes ending W1, M1 or X switch this off and tax each week or month on its own, so the figure can be wrong for a while, as the emergency codes page warns.
New Zealand works the other way round: you choose your own code on a tax code declaration. M is for your main income, and a student loan borrower picks a code that adds repayments, according to Inland Revenue. Give your employer no code and tax comes off at the non-notified rate of 45%, plus the ACC earners' levy, under the employer rules.
The take-home pay calculator assumes the standard code for the whole year, so a month on an emergency code can differ from its monthly figure.