Marriage Dispensations were issued if there was a question about a couple seeking to get married in a Catholic church by a priest. Then the matter would go before the bishop of the diocese. At that point a dispensation (permission) would or would not be granted for the marriage to be performed. In essence, a dispensation was an exception to what was normally forbidden by church law.
These are usually kept at the diocese offices, not being part of the parish registers. The way you know a dispensation was requested is that it is noted in the parish register. These records are usually considered somewhat sensitive, with many dioceses restricting who can and cannot study them. A call to the diocese archives should clarify their policy.
From a genealogist perspective, dispensations are tame. Reasons for a dispensation range from the couple being distantly related (consanguinity), an issue with the reading of the marriage banns in church or bringing a common-law marriage into a church marriage. From my experience, the most important ones are when a Roman Catholic wants to marry a non-Catholic by a priest. This generates an important record. I see Catholics marrying Protestants all the time, and the first thing I do is look for a dispensation. Again, from my experience, information about the Protestant may not include personal details. The good information, such as birth place and parent’s names will probably be on the Catholic party.
A good online article seeking to clarify some of the misconceptions about dispensations is Dan MacDonald’s “Some Notes on Marriage Dispensations in Roman Catholic Marriage Records”: www.brikwall.com/marrdispensations.html
These are fascinating records, and worth your time to pursue. They may be the only place you will locate the birth place in Ireland and the parent’s names for your immigrant ancestor.