Hello,
Ireland has 1, 2 and even several 3 digit area codes (without the starting zero). Generally, for landlines area code + local number total 8 digits. For example, Dublin has area code 1 so local numbers are 7 digit (1 ??? ????). Limerick and most other cities have 2 digit area codes so the local number is 6 digit.
Cellular numbers are 9 digit and have the same format all over Ireland: 8? ??? ????. No area codes are required for cellular numbers.
The starting zero you mentioned is used for dialing within Ireland only.