In order to work legally in Ireland you require a PPS (Personal Public Service) number. This is also known as a PPSN.
PPSN registration is all done online.
Here are the steps to get a PPSN / PPS Number:
You can register for a PPSN here: https://services.mywelfare.ie/en/topics/identity-services/personal-public-service-pps-number/.
You will need to register for a MyGovID account first: Click here for MyGovID
This website page contains a list of the documentation you require: https://www.gov.ie/en/service/12e6de-get-a-personal-public-service-pps-number/
* Make sure to select ‘STARTING EMPLOYMENT’ as your reason for needing a PPSN, and not ‘STARTING A COURSE IN A SCHOOL OR COLLEGE’.