I was reading that you pay national insurance until you reach retirement age. But when I got to 60 my employer said that after this age I did not need to pay NI and they stopped the contribution. Is this correct?
You pay national insurance contributions until your state pension age.
If you are a female born before April 1950, you will have stopped paying national Insurance at the age of 60, and this would be the same even if you continued to work.