If your employer still insists on a fit note for the first seven days a private note may be issued – this is not covered by the NHS and a fee will be payable.
After seven days, you will need an NHS fit note.
If you have been in hospital (e.g. for an operation), it should be issued by the hospital for the expected length of recovery. See DWP guidelines here.
If you are being treated by a GP then they will issue the fit note.
For information on claiming benefits due to illness you can look here.