Val,
You have sent a link to a list of abstracts from 1904, but you haven’t said what your ancestors name was, so I don’t know which one to look at. However, I can at least confirm these are just abstracts. So that’s a short summary used to create an index to the probate files for that year.
Let’s take as a random example: Joseph Walsh 20th June Administration of the estate of Joseph Walsh late of 1 Commercial St, Navan, Co Meath. Jeweller who died 24th Oct 1903 granted at Dublin to Rose Walsh the widow. Effects £272 etc.
The word “administration” tells you he died intestate, ie no will; though he died in 1903, probate wasn’t granted till 20th June 1904. The year probate was granted is the year the file is indexed under, not the year of death; Rose Walsh was appointed executrix. In this case she probably inherited most of the estate (unless he had dependent children) but you should not assume that the executor received anything, the executor could often be a family friend and might not be mentioned in the will at all; £272 was the total value of the estate, not necessarily what Rose received.
If it says “probate of the will” then there was a will. Likewise you sometimes see “administration (with will)” which is a case where no executors had been appointed or if they were, they were dead or no longer in contact with the family, and so the probate court had to appoint an executor who could administer the estate.
Probate could be granted at a number of locations around Ireland. Dublin was known as the Principal Registry but as you can see from that 1904 page, there were registries in Waterford, Limerick, Tuam etc.
If ordering a copy of a probate file, you need to mention the name, date of death, date probate was granted, place probate was granted and whether it was an administration or will case type. A lot of pre 1900 probate files were lost in the 1922 fire but in some cases transcripts exist, and in some others a duplicate copy of the original will itself has survived – often held by a solicitor - and may now be held in the National Archives (or PRONI for Northern Ireland). You can search the NAS site using this link. Try “Calendar of Wills” plus the year(s) you are interested in.
https://www.nationalarchives.ie/search-the-online-catalogue/simple/#!/
This site contains a very helpful guide to where wills were kept and what happened to them. It also has the calendars (indexes) to the post 1922 wills:
http://www.cigo.ie/pages/guide-to-wills-and-administrations/
If you are searching for a will and don’t know the date of probate, it’s wise to search several years after the death. I have known cases where probate wasn’t granted till 25 years later (often farmers who didn’t bother to get probate at all, and it eventually comes to light years later when someone tries to sell the property or when the inheritor dies).
For a very detailed explanation about leases & land ownership etc, try: “Researching Farming Ancestors in Ireland” by Dr William Roulston (publ 2021).