The answer to this basically depends on what time period you're looking at. The Celts, much like the various Germanic peoples throughout Europe, lived at first in pastoral societies that underwent enormous change through contact with the Romans.

Prior to, and immediately following Roman contact, the Celts had developed a pastoral economy in which oxen, sheep and pigs were raised on the plentiful suitable land. This pastoral activity was supplemented by hunting, fishing, beekeeping and some farming (Arthur Birnie, Economic History of the British Isles, p. 14). Although some primary sources, such as writings by Caesar, suggest that farming was fairly central to the British economy of the time, archaeological evidence does not support this. For example, the Celts made use of fairly time consuming and primitive farming tools and techniques.

In these pastoral economies, when farming was done it usually had some basis in the previously existing communal societal organisation. For the Celts, this took the form of a rundale system. Although social stratification had taken place, and there were privileges associated with social position, within each strata there was a degree of equality. In each tribe/clan, leading figures would identify the land to be tilled that year and it would be divided up amongst the constituent groups and farmed communally. Each year or so the land would be reallocated which prevented substantial build-up of private wealth. However, there is evidence (particularly in the disparity of grave sites and primary source references to Celtic law) to suggest that private ownership of animal herds and other similar property probably existed (Perry Anderson, Passages from Antiquity to Feudalism, London, 1996, pp.107-8).

However, by the First Century, contact with the Romans had substantially changed Celtic social arrangements (as with the various Germanic tribes). Trading for luxury goods, cattle raids and pressure to produce more agrarian output for trade with the Romans led to an increasingly stratified society. Land began to be distributed directly to individuals rather than ploughed communally and eventually reallocations became infrequent or ceased entirely. Abundant land and seasonal raids led to the development of hereditary leading councils and quasi-royal lineages with cadres of trained warriors which came to dominate Celtic politics (Thompson, The Early Germans, pp.48-60). As these groups cemented their power, hereditary private ownership of land came to be the norm along the Roman frontier among both the Germanic tribes and the Celtic peoples who had been displaced by the migrations.

Some of this is also covered in a great book by Chris Wickham called the Inheritance of Rome which clarifies a lot of stuff about social arrangements in the wake of the collapsing Roman Empire. It debunks a whole bunch of myths and outdated history (including some of Anderson's) and is a great resource to find out more.

It's problematic to talk about the "Celts" as a single monolithic civilization; the term only really refers to people who spoke Celtic languages across Europe and Asia and does not imply similarities in political or social structures. Because of this, we can't assume that patterns of landholding would be the same or even vaguely similar from Ireland to Gaul to Britain to Iberia, Southern Germany, Anatolia etc...

So to answer your question a bit: in medieval Ireland the concept of property did exist, but it didn't really resemble our contemporary notion of private property. Property was not owned by individuals but was held by the fine - the extended family unit. In early and medieval Ireland, the individual was always subordinate to the extended family unit, even in legal conflicts; an offence committed by one man against another was not seen as a matter to be settled between two individuals but one that concerned the fine of both men.

Although land was held by the fine, it was distributed to individual nuclear families that made up the extended family unit. Two strategies were employed to avoid intra-group conflict arising from the potentially unfair distribution of land: the first was that each family would be distributed a variety of terrain to ensure that they could be economically self-sufficient to an extent - each family would be distributed arable land for agriculture, peat bog for fuel, woodland for swine raising, building materials and food and so on. The second strategy was the almost constant redistribution of land within the fine's territory to its members. The frequency and timing of this strategy largely depended on the region of Ireland that you're looking at, but land was redistributed to different families to ensure that one or several did not hog the best farmland at the expense of their fellow fine members. This redistribution might have happened yearly or every couple of years and usually took place on a religious or harvest festival.

So in Ireland, there was a concept of property but it was one that did not resemble modern conceptions of private property. Land was held in common by the extended family group and was divided between its constituent members to ensure an equality in resources available and to avoid conflict within the group over access to resources.