(This answer is largely copied from text I've used for a class I've taught at SCA events. I think I've removed all the SCA references and injokes.)

Food history of Ireland still isn't a very developed area of work. This is mostly because there are very few written sources from which to work. There are no Irish cookery books before the 17th century, and the earliest real reference to Irish cuisine in the form of a recipe is from Ouverture de Cuisine, published in Liege in 1604. That work contains references to roasting a leg of mutton ‘in the Irish style’, and also recipes to garnish a duck and make a peeled veal head, both marked as Irish. But that's pretty late.

There are, however, other ways to get at information about Irish medieval food - texts, physical remnants, descriptions by visitors to the island, and the recipes that did appear in the 17th century. For purposes of this answer I’ll be looking principally at texts - in particular, law texts - and physical remnants. It’s hard for non-natives to get from Irish spelling to correct pronunciation, so I’ve provided some guidance on that in the text.

The principal texts which from which we can get information are the 7th century Críth Gablach (crih’ gawb-lock), which deals with rank and privilege, the 8th century Bretha Comaithchesa (bre-ha co-moh’-ke-sa), which deals with judgements concerning neighbours and trespass, and the Cáin Aicille (caw-in ock-il-eh), the law of clientship and patronage, also from the 8th century. It's worth bearing in mind here that these texts are primarily lists of precedent and accounts of judgements, rather than laws which were set forth in principle.

Dairy and grain were more widely eaten than meat. However, meat animals feature more in the law texts, so we’ll deal with that before anything else.

Far and away the most important animals in medieval Ireland were cattle. They were traded, raided for, milked, had blood drawn, were slaughtered and eaten in every part. Beef and what we would call veal were therefore important meats, particularly for the nobility. Bleeding of cattle was also practiced, mostly in times when other food was scarce, and bone marrow was extracted as well - every part of the slaughtered animal was used.

Mutton was eaten, but sheep were kept much more for wool. However, there’s some evidence to suggest that mutton was a traditional wedding food. By the later end of the medieval period, this had probably changed by quite a bit, as evidenced by the mutton recipe in the 1604 French text.

Pigs, in contrast, were kept solely for meat, and were widely eaten. The medieval Irish pig was small, hairy, and long-legged, and probably quite lean in comparison to modern pigs. The regard in which pigs were held is illustrated by there being a penalty in one law-text for disturbing a sow by imitating the squealing of her piglets. Pigs were fed on a wide variety of foods, and particularly acorns in autumn - but were not usually let loose in woodlands as they were in other parts of Europe, as their trespass onto other people’s land was held to be more damaging than that of other animals. Suckling pigs were sometimes eaten, but by and large, pigs were full grown and fattened before slaughter, and these fat pigs would also provide pork lard.

There are laws that concern chickens and geese - all kept for both eggs and food. Goose eggs were notably more valued than hen eggs, and there’s at least one mention in the Middle Irish story Fled Dúin (Fled Doo-in) of a King of Ulster being deeply insulted when he received a hen egg at a feast while other kings had goose eggs. Ducks, peculiarly, are not mentioned in law texts at all, and while doves were known, their use as a domestic bird seems to have arrived with the Normans.

We know the name of a law text concerning the sea and fishing - Muirbretha (Mwir-bre-ha), but the actual text is lost. References from elsewhere make it clear that salmon, trout, and eels were eaten, but that’s almost all the information that’s available.

Cereals don’t feature as much in the laws as livestock - presumably because grain was less likely to wander onto the neighbours’ land or injure them. However, the comparative value of grain was a matter which which the law was concerned, and so the 8th century Bretha Déin Chécht (Bre-ha day-in kekt, approximately) gives this order of precedence in descending order: bread-wheat, rye, spelt-wheat, two-row barley, emmer wheat, six-row barley, and oats. Several of the terms used in the list are words whose finer distinctions have been lost - so two-row and six-row are guesses, and the identity of rúadán (roo-ah-dawn, emmer wheat in my list) has been extensively debated. The one thing it is definitely not is buckwheat, because that wasn’t introduced to Europe until the 13th century, and isn’t a grain.

Bread and porridge were both made with all forms of grain, even the wheat (wheat porridge is specifically mentioned as the entitlement of the children of kings). Medieval breeds of wheat did not grow well in Ireland’s damp climate, though, so there was scarcity as well as quality involved. This influences the rest of the list as well - the grains make less and less good porridges and breads until you get to oats, but oats grew everywhere, more or less, and were therefore the food of commoners, even if they made better food than some of the more prestigious grains. Beer was made principally from barley, rather than any other grain.

The justice of a king, it is known, is responsible for the wealth of milk that come from the cows in his kingdom. If the cows give much milk, the king is just. If the cows are dry, why, the king must be unjust. Cattle were probably more important in medieval Ireland for milk than for meat on a day to day basis, and the production of butter, cheese, whey, and other goods were very important in the Irish diet.

Sheep’s milk was also used, but was considered rather inferior to that of cows and goats. Most of it was therefore made into cheese, which was a very important foodstuff, and much appreciated by the Irish. This is one of the notable differences between medieval Irish and medieval English food; the English generally considered cheese of any kind to be something to be eaten in desperation, not a good food in its own right. Goats were kept mostly for milk, according to the textual evidence, although goat-skins were used as well.

Bees are a topic much covered in the law texts, principally because it was completely impossible to stop them from trespassing. Trespass - as noted with pigs - is a major part of Irish law pertaining to agriculture, and the Bretha Comaithchesa concentrates mostly on this. The rule for bees was that when someone got a hive, they gave away a swarm each year to their neighbours, the closest neighbour first, so that after a few years, everyone had a few hives, and there was no need to worry about trespass, because all the bees were assumed to go everywhere. Honey was valued greatly, though, being the main source of sweetening apart from fruit, and for the making of mead - a more prestigious drink than beer.

Other crops are much less referred to. We know of definitive references to peas and to beans (probably broad beans), and after that, things get a little fuzzy. Cainnenn (caw-in-nen) is mentioned frequently, but never with enough clarity to decide whether it refers to onions or garlic. Imus (ih-mush) is probably celery, and borrlus (bor-lush) is probably leek. Braiseach is definitely something in the brassica family, most probably cabbage or something close to it - but medieval brassicas were not as clearly distinguished as our modern ones in any case.

Most peculiarly, root crops are very little referred to. There are a few limited references to cerrbacán, which might be carrot or skirrit, but otherwise they’re just ‘roots’ without anything to distinguish them. Considering the importance of the turnip and the potato in later Irish diets, this is an oddity.

The only herb that’s reliably identifiable from Irish texts is chives, and while there are mentions of a number more, there is nothing from which we can work out their actual identities. Indeed, some of the names are tied directly to the rank of those entitled to eat them, so that we can read about ríglus (reeg-lush), tarblus (tawrb-lush), and aithlechlus (aw-ith-lekh-lus), translated approximately as “king’s herb”, “bull’s herb” (that is, for the cattle-owning people), and “plebian herb” - but we have no idea what plants they were.

Among fruit, the apple is definitively known. The plum is fairly reliably known (and there’s archaeological evidence as well). Pears seem to have been a Norman introduction, and even in the warmer parts of the medieval climate, there is nothing to show that grapes grown in Ireland (despite anything Bede may say on the matter). It’s possible that they could have been grown on the south coast (and I’ve seen feral grape vines there myself in the 1980s), but there’s no evidence in text or archaeology.