It's unclear from your post whether you mean 'begin the process' of applying for the State pension or applying for the PSC!
I assume that you know that you begin the process of applying for the State Old Age pension (there are two types) by filling in an application form!
As for the need for a PSC, some years ago the government tried to make the PSC compulsory, but it later dropped that requirement after some criticism from various bodies including the Data Commissioner, the ICCL, the opposition, the usual Irish Times journalists (!) and the Ombudsman.
So it's no longer compulsory, however it's a useful thing to have, and it's advisable to get one so you can have your free travel pass incorporated into it when you turn 66! See above for how.
You should get an Application Form in the post from the Dept of Soc Prot about 3 months before your 66th birthday . If not, you can download the form from the website Welfare.ie. Alternatively you can get a form in your local Intreo office.