The Old Age Pension was introduced in the UK & Ireland in 1909. To qualify you had to be 70 or over. Proof of age was required. However since birth certificates were only introduced in Ireland in 1864 (in contrast to 1837 in England), many applicants couldn’t produce one. So in Ireland other documents were acceptable, eg baptismal certificates, marriage certificates (if they established your age satisfactorily) and military discharge documents which normally contained the person's age. However if you had none of those, another approach was to check the 1851 (and sometimes the 1841) census. Obviously if a pension applicant in say 1917 was 4 or older in the 1851 census, then they’d be over 70 and so eligible. So there was a system whereby applicants without proof of age could complete a form stating where they were living (or thought they were living) in 1851, and where relevant, in 1841. This information would then be sent to the Public Record Office in Dublin and checked against the censuses. If their family was found in the relevant census and the age accurate, then they’d get their pension.
The pension was non-contributory, with the cost being borne by taxpayers generally. It was enacted in 1908 and was to pay a weekly pension of 5s a week (7s 6d for married couples) with effect from 1 January 1909. The level of benefit was deliberately set low to encourage workers to go on making their own provision for retirement. In order to be eligible, claimants had to have an income less than £31. 10s. a year, and also had to pass a 'character test'; only those with a 'good character' could receive the pensions. Claimants also had to have been resident in Great Britain and Ireland for at least twenty years to be eligible, and those who had not worked habitually were also not eligible.
In the document you have found Catherine Mulvey was applying for a pension in September 1921 (indicating she had probably recently reached 70). She told the authorities that in 1851 she believed she was living in one of 3 townlands ie Arderry, Drumlea or Drumreilly. She gives her parents' names. The clerk in Dublin evidently searched the 1851 census (taken 30th March 1851) and found an entry to confirm Catherine’s information. It is not clear in which of the 3 townlands the family were living. The notes (as I read them) indicate Catherine's parents married in 1840. In addition to finding Catherine in the 1851 census, there were siblings Owen, Jane, Margaret, Mary, Ann & James. Plus Thomas who had died. (The 1841 & 1851 censuses asked for details of family members who had died or were not at home on census night. Information dropped after those 2 censuses). Some of the other notes are to do with the search fee. Catherine appears to have sent 3 shillings (3/=) but the appropriate fee was only 2/6 ie 2 shillings and sixpence. Or half a crown, as I recall it. So she was due a refund of 6d (sixpence), which was sent on 30.9.1920.